Advances in Reliability

Preface The area of Reliability has become a very important and an active area of research. This is clearly evident fro...

Author: N. Balakrishnan | C. R. Rao

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Preface

The area of Reliability has become a very important and an active area of research. This is clearly evident from the large body of literature that has been developed in the form of books, volumes and research papers since 1988 when the previous Handbook of Statistics on this area was prepared by P. R. Krishnaiah and C. R. Rao. This is the reason we felt that this is indeed a right time to dedicate another volume in the Handbook of Statistics series to highlight some recent advances in the area of Reliability. With this purpose in mind, we solicited articles from leading experts working in the area of Reliability from both academia and industry. This, in our opinion, has resulted in a volume with a nice blend of articles (33 in total) dealing with theoretical, methodological and applied issues in Reliability. For the convenience of readers, we have divided this volume into 13 parts as follows: I Reliability Models II Life Distributions III Reliability Properties IV Reliability Systems V Progressive Censoring VI Analysis for Repairable Systems VII Analysis for Masked Data VIII Analysis for Warranty Data IX Accelerated Testing X Destructive Testing XI Test Plans XII Software Reliability XIII Inferential Methods We hope that this broad coverage of the area of Reliability will not only provide the readers with a general overview of the area but also explain to them what the current state is in each of the topics listed above. We express our sincere thanks to all the authors for their fine contributions and for helping us in bringing out this volume in a timely manner. Our special thanks go to Ms. Nicolette van Dijk for taking a keen interest in this project and also for helping us with the final production of this volume. N. Balakrishnan C. R. Rao

Contributors

J. A. Achcar, ICMC, University of $8o Paulo, C.P. 668, 13560-970, Säo Carlos, SP, BraziI, e-mail: [email protected] (Ch. 29) R. Aggarwala, Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4, e-mail." [email protected] (Ch. 13) R. Agrawal, GE Corporate Audit Staff, Fairfield, CT 06432-1008, USA, e-mail: [email protected] (Ch. 27) P. A. Akersten, Center for Dependability and Maintenance, Luleä University of Technology, Luleä, Sweden, e-mail: [email protected] (Ch. 16) E. K. AL-Hussaini, Department of Mathematics, University of Assiut, Assiut 71516, Egypt, e-mail." [email protected] (Ch. 5) S. Aki, Division of Mathematical Science, OsaÆa University, Graduate School of Engineering Science, Toyonaka, Osaka 560-8531, Japan, e-mail: [email protected] (Ch. 11) M. Asadi, Department of Statistics, University of Isfahan, Isfahan 81744, Iran, e-mail: [email protected] (Ch. 7) N. Balakrishnan, Department of Mathematics and Statistics, McMaster University, Hamilon, Ontario, Canada L8S 4Kl, e-mail: [email protected]. mcmaster.ca (Chs. 1, 14, 23) U. Balasooriya, Department of Statistics and Applied Probability, National University of Singapore, Lower Kent Ridge Road, Singapore 119260, e-mail: [email protected] (Ch. 15) M. Banerjee, Center for Health Care Effectiveness Research, Wayne State University, Detroit, MI 48201, USA, e-mail." [email protected] (Ch. 19) A. P. Basu, Department of Statistics, University of Missouri at Columbia, Columbia, MO 65211-0001, USA, e-mail: [email protected] (Ch. 2) S. Basu, Division of Statistics, Northern Illinois University, DeKalb, IL 601152854, USA, e-mail: [email protected] or [email protected] (Ch. 19) B. Bergman, Division of Total Quality Management, Chalmers University of Technology, Gothenburg, Sweden, e-mail." [email protected] (Ch. 16) W. R. Blischke, Emeritus Professor, Department of Information and Operations Management, University of Southern California, Los Angeles, CA 90089-1421, USA, e-mail: [email protected] (Ch. 20)

xix

xx

Contributors

A. Chatterjee, Department of Statistics, Burdwan University, Burdwan 713104, W. Bengal, India, e-mail." [email protected] or [email protected] (Ch. 4) E. Cramer, Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany, e-mail: [email protected] (Ch. 12) N. Doganaksoy, GE Corporate Research and Development, K14C35, Niskayuna, N Y 12309, USA, e-mail: [email protected] (Chs. 26, 27) N. Ebrahimi, Division of Statistics, Northern Illinois University, DeKalb, IL 60115-2854, USA, e-mail: [email protected] (Ch. 31) B. J. Flehingert, IBM Research Division, Mathematieal Sciences Department, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA (Ch. 18) S. K. Ghosh, Department of Statisties, University of North Carolina, Raleigh, NC 27695, USA, e-mail." [email protected] (Ch. 31) E. Gouno, Department of Applied Statistics (SABRES), University of South Brittany, Rue Yves Mainguy, Tohannic, F56 000 Vannes, France, e-mail: [email protected] (Ch. 23) G. Y. Hong, Institute of Information and Mathematical Sciences, Massey University, Auckland, New Zealand, e-mail: [email protected] (Ch. 28) K. Hussein, University of Wisconsin, Department of Mathematics, 400 University Drive, Fond du Lac, WI 54935 USA, e-mail: [email protected] (Ch. 33) R. A. Johnson, Department of Statistics, University of Wisconsin, 1210 W. Dayton Street, Madison, WI 53706-1685, USA, e-mail: [email protected] (Ch. 24) U. Kamps, Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany, e-mail: [email protected] (Ch. 12) N. Kannan, Department of Mathematics and Statistics, University of Texas at Sah Antonio, San Antonio, TX 78249, USA, e-mail." [email protected] (Ch. 14) Md. R. Karim, Department of Statistics, University of Rajshahi, Rajshahi- 6205, Bangladesh, e-mail: [email protected] (Ch. 21) L. B. Klebanov, Faculty of Mathematics and Mechanics, Division of Statistics and Probability, St. Petersburg State University, St. Petersburg 198904, Russia, e-mail: [email protected] (Ch. 9) B. Klefsjö, Center for Dependability and Maintenance, Lule{t University of Technology, Luleä, Sweden, e-mail." [email protected] (Ch. 16) K. B. Kulasekera, Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA, e-mail: [email protected] (Ch. 30) C. D. Lai, Institute of Information Sciences and Technology, Massey University, Palmerston North, New Zealand, e-mail: [email protected] (Ch. 3) N. Limnios, Département Génie Informatique, Division Mathématiques Appliquées, Université de Technologie de Compiègne, B.P. 20 529, 60205 Compiègne Cedex, France, e-mail: [email protected] (Ch. 1) W. Lu, Department of Statistics, University of Wisconsin, 1210 W. Dayton Street, Madison, WI 53706-1685, USA, e-mail: [email protected] (Ch. 24) M. Mazumdar, Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA, e-mail: [email protected] (Ch. 25)

Contributors

xxi

G. C. McDonald, Director, Enterprise Systems Lab, General Motors Research and Development Center, MC #480-106-359, 30500 Mound Road, Warten, MI 48090-9055, USA, e-mail." [email protected] (Ch. 17) J. Mi, Department of Statistics, Florida International University, University Park, Miami, FL 33199, USA, e-mail: [email protected] (Ch. 8) N. A. Mokhlis, Department of Mathematics, Faculty of Science, Ain-Shams University, Cairo, Egypt, e-mail: [email protected] (Ch. I0) S. P. Mukherjee, Department of Statisties, Calcutta University, Calcutta 700019, W. Bengal, India, e-mail: [email protected] or [email protected] (Ch. 4) D. N. P. Murthy, Department of Mechanical Engineering, University of Queensland, Brisbane, Queensland 4072, Australia, e-mail: [email protected] (Chs. 3, 20) P. R. Nelson, Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA, e-mail: [email protected] (Ch. 30) W. Nelson, Consultant, 739 Huntingdon Drive, Sehenectady, N Y 12309, USA, e-mail: [email protected] (Ch. 22) S. Panchapakesan, Department of Mathematics, Southern Illinois University at Carbondale, Carbondale, IL 62901-4408, USA, e-mail." [email protected] (Ch. 33) C. Papadopoulos, D@artement Génie Informatique, Division Mathématiques Appliquées, Université de Technologie de Compiègne, B.P. 20 529, 60205 Compiègne Cedex, France, e-mail." [email protected] (Ch. 1) J. Rajgopal, Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA, e-mail: [email protected] (Ch. 25) B. Reiser, Department of Statistics, University of Haifa, Haifa, Israel (Ch. 18) S. E. Rigdon, Department of Mathematics and Statistics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026-1653, USA, e-mail: [email protected] (Ch. 2) A. Sen, Department of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USA, e-mail: [email protected] (Ch. 19) M. Shaked, Department of Mathematics, University of Arizona, Tuscon, AZ 85721, USA, e-mail: [email protected] (Ch. 6) D. N. Shanbhag, Department of Probability and Statistics, University of Sheffield, Sheffield, $3 7RH, England, UK, e-mail." [email protected] (Ch. 7) F. Spizzichino, Dipartimento di Matematica, Universita degli Studi di Roma ùLa Sapienza", Piazzale Aldo Moro, 2, 00185 Roma, Italy, e-mail." [email protected] (Ch. 6) J. Stein, G.E. Corporate Research and Development, Building K1-4C27A, One Research Circle, Niskayuna, N Y 12309, USA, e-mail: [email protected] (Ch. 26) K. S. Sultan, Department of Mathematics, AI-Azhar University, Nasr City, Cairo 11884, Egypt (Ch. 5)

xxii

Conlributors

K. Suzuki, Department of Systems Engineering, The University of ElectroCommunications, 1-5-1 Chofugaoka, Chofu-city, Tokyo 182-8585, Japan, e-mail: [email protected] (Ch. 21) G. J. Szekely, Department of Mathematics and Statistics, Math Science Building, Bowling Green State University, Bowling Green, OH 43403-0221, USA, e-mail." [email protected] (Ch. 9) L. Wang, Graduate School of Information Systems, University of ElectroCommunications, Tokyo 182-8585, Japan, e-mail: [email protected] (Ch. 21) M. Xie, Department of IndustriaI and Systems Engineering, The National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, e-mail: [email protected] or [email protected] (Chs. 3, 28) E. Yashchin, IBM Research Division, Mathematical Sciences Department, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, N Y 10598, USA, e-mail: yashchi@us,ibm.com (Ch. 18) S. Zacks, Department of Mathematical Sciences, Binghamton University, Binghamton, N Y 13902-6000, USA, e-mail: [email protected] (Ch. 32)

N. Balakrishnan and C. R. Rao, eds., Handbookof Statistics, Vol. 20 © 200i Elsevier Science B.V. All rights reserved.

| 1

Basic Probabilistic M o d e l s in Reliability

N. Balakrishnan, N. L i m n i o s and C. Papadopoulos

Notation SMP EMC r.v,

RE

R(t)

A(t) A

M(t) MTTF MTTR MUT MDT MTBF

N(t) ~.(t) Qij(t) p8

H,(O ~ij(t)

semi-Markov process embedded M a r k o v chain r a n d o m variable relative error reliability function pointwise availability function limit availability maintainability function system failure rate function mean time to failure mean time to repair mean up time mean down time mean time between failures number o f j u m p s of the semi-Markov process in the time interval

(0,t] number of visits of the semi-Markov process into the state i in the time interval (0, t] semi-Markov kernel: discrete state space case; i E E, j E E, t E IP,+ transition function of the M a r k o v chain (Jn): discrete time case distribution function of sojourn time in the state i, i E E M a r k o v renewal function: discrete state space case;

i EE, j EE, t E IR+ #U #u mi

«(x,; t E ,r) Q1 -x Q2 Q(,)

initial law mean hitting time of the SMP into the state j, starting in state i mean hitting time of the E M C into the state j, starting in state i mean first jump time under IPi or mean sojourn time in state i the a-algebra generated by the family of r a n d o m variables (Xt; t E I) Stieltjes convolution of two semi-Markov kernels on E nth fold Stieltjes convolution of the semi-Markov kernel Q, n E N Stieltjes convolution product

2 «

a.s. N N*

IP, IR+ lA

N. Balakrishnan, N. Limnios and C. Papadopoulos

matrix Stieltjes convolution product almost surely the set of natural numbers: {0, 1 , 2 , . . } the set of positive natural numbers: { 1 , 2 , . . } the set of real numbers the set of nonnegative real numbers: [0, ec) indicator (of characteristic) function of a subset A; lA(x)=

l(x)

d ___+ a.s. _____+

N(0, 1)

ifxŒA ifx~A

the Heavyside function on IR;

l(x)= 1s~/"

{1 0

I1

0

ifx_>0

ifx_ 1 and all i0, il, . . . , in E E, we have: 1. IP(Xo = io,X1 = i l , . . . ,Xn 1 = in-l,Xn = in) = c t ( i o ) P ( i o , i l ) . . . P ( i n - l , i n ) ;

2. 1P(Xù+I -----i l , . . . ,Xn+k-~ = ik-l,Xn+k = ikIXn = io) = P(io, il)...P(ik-1, ik); 3. ]P(Xn+m = jlXm

=

i) = IP(Xn = j[Xo = i) = p n ( i , j ) .

4

N. BaIakrishnan, N. Limnios and C. Papadopoulos

PROPOSITION 2.2. 1. The sojourn time o f the system into the state i E E is a geometric r.v. with parameter P ( i , j ) . p(ij) 2. The probability that the system enters stare j when it leaves stare i is' 1-p(i,i)"

The above two propositions allow us to simulate by M o n t e Carlo m e t h o d s a M a r k o v chain. EXAMPLE 2.1. (Binary c o m p o n e n t ) Consider a binary c o m p o n e n t (or system) having an up (functioning) state and a down (failure) state that we denote by 1 and 0, respectively. At time n = 0 the c o m p o n e n t starts functioning until its failure at a r a n d o m time, S1 say, where it is replaced instantaneously by an identical one which lasts a r a n d o m time, $2 say, and so on. The lifetime distribution of the c o m p o n e n t s is 0 = (On,n E N*), i.e. IP(S1 = n ) = 0 , The r.v.'s U1, U 2 , . . defined by: U~ = Sn - Sn 1, (n _> 1, So = 0), are i.i.d, and represent the lifetimes of the successive components. Assume now that the distribution 0 is the geometric one with p a r a m e t e r p. F o r a time n >_ 0, define the r.v. X~ with values in E = {0, 1}. The event {X~ = 1} means that the c o m p o n e n t is functioning at time n and the event {X~ = 0} means that the c o m p o n e n t is failed at time n. Then it is clear that the stochastic process X = (Xù, n >_ 0) is a M a r k o v chain with state space E, transition matrix P

~(~~ ~) q

1-q

and initial distribution (a, 1 - ~) (c~ E [0, 1]). After some calculus we obtain the following spectral representation for the powers of the transition matrix P: pn =

1- p q

p 1

q

1 --p+q

-~ p+q

-q

(3) The state probability vector is (4)

P(n) = (P1 (n),P2(n)) , Pl(n) -

q P+q

~-

(1 - p

-

q)n

(p~ - q(1 - ~)) .

(5)

P+q

2.1.2. Recurrent and transient states A state of a M a r k o v chain m a y be either transient or recurrent. This is a f u n d a m e n t a l notion in the study of M a r k o v chains. L e t j be a fixed stare and the r.v. S j, s J , . . , represent the times of the 1st, 2nd,...

return to stare j. Define also U~ - S~ - S~_I, J n = 1,2,..

(take S° = So), to be the

B a s i c p r o b a b i l i s t i c m o d e l s in reliability

5

times between two consecutive returns to state j, which are called recurrence times of state j and

Nr:=~ l{xk=j}

,

(6)

k=l

N/~.:= ~

l(X k ~=i,Xk=j} ,

(7)

k=l

Nj := ev~ ,

(8)

Nij:=N,f

(9)

,

p,y := ]Pi(S{ < oo) ,

(10)

B/j:= IEiS{ .

(11)

DEFINITION 2.2. A state i is called a recurrent s t a t e i f Pii = 1. On the other hand, if Pii < 1, then i is called a transient stare. I f i is a recurrent state lhen either ~ii < oo, and it is called recurrent positive state or #ii = OO, and it is called null recurrent state. PROPOSlTION 2.3.

1. A state i E E is transient iff ( i f and only if) lPi(Ni = + o o ) = 0 or / y ~ n P " ( i , i ) < oo. 2. A state i E E is recurrent ifflPi(Ni = +oo) = 1 or iff ~ n P n ( i , i ) = O0.

PROPOSITION 2.4. I f i and j are two recurrent states, then N n - -

n N n

--

a., l --~ - - » [lii

ij ~.s. _ / . . ~

a s 17 ---+

1

--+ v ~ , 1 , - - ,

n

O0

(12)

»

asn--+oc

.

(13)

#ii

2.1.3. Stationary probability and asymptotic behavior DEFINITION 2.3. A probability distribution on E, is called stationary or invariant with respeet to the M a r k o v chain X (or to the transition probability P ) , if f o r all j E E, we have Z

re(i)P(i,j) = ~(j') .

(14)

lEE

Eq. (14) can be written in matrix form as reP : re .

PROPOSlTION 2.5. For every recurrent state i, we have: re(i) = l /#ii. For each state i, deßne di = g.c.d. {n : n > 1,P"(i,i) > 0}.

6


DEFINITION 2.4. A state i E E is said to be periodic if di > 1, and aperiodic if di = 1. DEFINITION 2.5. A recurrent positive and aperiodic state is called an ergodic state. An irreducible M a r k o v chain with all its states aperiodic is called an ergodic Markov chain. PROPOSITION 2.6 (Ergodic t h e o r e m for M a r k o v chains). For an ergodic M a r k o v chain we have P~(i,j) --+ re(j),

as n --+ oc .

PROPOSITION 2.7. I f E is finite and the chain is irreducible and aperiodic, then exists and pn converges toward FI = l~z with an exponential rate.

2.2. Reliability in countable time 2.2.1. Introduction The reliability of a system can be studied in countable time too; for example in N instead of IR+. This lies in its easy formulation and calculus. This countable formulation that can be used as a first approach, m a y also be, in m a n y cases, sufficient. Let us consider a c o m p o n e n t (of a system) that is observed in times n = 0, 1 , 2 , . . and suppose that at time n it occupies stare x a m o n g a n u m b e r of possible stares given by its stare space. In the time interval (n, n + 1), where the c o m p o n e n t is not observed, it can change or not its state with probabilities p and q = 1 - p , respectively. We can then define the "failure rate", the reliability, etc. Let us define the N - v a l u e d r.v. T which denotes the lifetime of the a b o v e c o m p o n e n t , then the failure rate is defined as follows: )o(n) := IP(T = n]T >_ n)

(15)

for any n E N. The sequence 2 = (2(n),n E N ) is a failure rate o f a c o m p o n e n t or a system if 0 _< 2(n) _< 1 and ~n>0 2(n) = +oo. Moreover, the probability of (ailure at time n _> 0 is

(]ln-1 f ( n ) = IP(T = n ) =

)

\ ~ . ö [ 1 - 2(i)]

)~(n)

(16)

with the convention that H ó (') = 1. The reliability at time n > 0 is given by

n i--0

R(n) = IP(r > n) = H [ x - 2(i)1 .

If R(m) > 0, we have

(17)

Basic probabilistic models in reliability f(n)

;(~)

-

R(~)

7

(18)

for all 0 < n < m.

2.2.2. Muhistate Markov systems Consider a c-order system (C, (p), where C - { 1 , . . ,c} stands for the set of its components (c E N*) and (p for its structure function. Ler S be the state space of the system and of its components. (There is no loss of generality by considering the same state space for all components and for the system.) For example S = {0, 1 , . . ,M}, where M is the perfect state of the components and of the system and 0 the complete failure. When M = 1 we say that the system is binary with 1 the working state and 0 the failure state. Let X i = (Xi, n E N) be an S-valued stochastic process describing the behavior of the component i, and ( X 2 , . . ,X~) be the SC-valued stochastic process. This last process describes jointly the states of the system components. Consider now a one-to-one mapping h : S c -+ E, where E = { 1 , . . , s}, s = # S « and the process Xn = h ( X 2 , . . ,X~). In general we take h to be the lexicographic order of the set S c. EXAMPLE 2.2. Consider a second-order binary system. We then have: c = 2, S = {0, 1}, S 2 = {(0,0),(0, 1), (1,0),(1, 1)} and E = {1,2,3,4}. If the components are independent and the above processes X i are Markov, then the process X is a Markov chain too. The following proposition gives a more precise description. PROPOSITION 2.8. Consider two independent Markov cha&s, X 1 and X 2 say, $1 and $2 valued with transition functions p1 and p2, respectively. Then the (X1,X 2) is an S 1 x S2-valued Markov chain with transition function P, with P((i,j), (k, g)) -p l ( i , k ) p 2 ( j , g ) , for all i,k E S 1 and j, g E S 2. The above proposition can be generalized for more than two processes. In the case of binary systems, we have to partition the whole stare space E into disjoint sets, U and D say, where U includes the working states (up stares) and D the failure states (down states), (i.e. U U D -- E, U N D = ~ and U ¢ (~, D ¢; (~.) The reliability-related indicators at time n > 0, become: • Reliability: R(n) = IP(Vr E [0, n] M N,Xv E U). • Availability (pointwise): A(n) = 1P(Xn E U). • Maintainability: M(n) -- 1 -1P(Vv E [0, n I N N,X~ c D). Consider now a binary multistate system (BMS) with state space E={1,..,s}, up states U = { 1 , . . , r } and down states D = { r ÷ l , . . . , s } , described by an E-valued Markov chain X, with transition probability matrix P and initial distribution vector et.

8


Let us consider the following partition of P and c~corresponding to the U and D sets:

p = (Ph

\

P21

=(
_ 1,

2 j ~ ~ «(J),

n = 0

A(n) = { Œ[L=0Pkl~,r, PROPOSITION 2.15.

R(n) = { PROPOSITION 2.16.

The reliability of an N H D T M C system is given by o~ "Fln l

U

1 Ilk=0 Pk L ,

n >_ 1,

The maintainability of an N H D T M C system is given by

n 1 D Pk 1~_~, n _> 1, A(n) = { 1 -- ~X2 Hk=0

1 - ~ j c D c~Ü),

n = 0 .

Basic probabilistic models in reliability

11

The M T T F and M T T R of an N H D T M C system are given by

PROPOSITION 2.17.

M T T F = Cq ( I X /

MTTR

= ~2 ( I

\

oe k D)

+Z~P~~=0 1~~

}~XAMPLE 2.5. Consider an absorbing N H D T M C matrices, for n = 0, 1 , 2 , . .

with transition probability

( (n+l)a , (n+l)a ~ 1_~) Pn =

1

1

(n+l)b0 (n+l)b0

1

~

j

,

where a > 1 and b > 1. The stare space partition is: U = {1,2} and D = {3}. Then, we have: 2(a + b) ~-I

A(n)=R(n)-

n[anbù_l ,

n> 1

and A(0) = R(O) = 1. M o r e o v e r , MTTF=I+

2b .1+~ [e~ - 1 ] a+b

.

3. Continuous time Markov chains and reliability

3.1. Definition This section deals with continuous time M a r k o v chains, i.e. I = IP,+. As in the previous section we will consider here a probability space (f~, ~ , IP) where we define an E-valued stochastic process X = (X(t), t c IP,+). The state space E here is at m o s t a countabte space. DEFINITION 3.1. The stochastic process X is a continuous time Markov chain

( C T M C ) if, for all j E E, and all t, h >_ O, we have Ip(X(t+h) =j]X(u),u < t) = Ip(X(t+h) =jIX(t)),

(a.s.) .

(25)

We consider here time h o m o g e n e o u s processes in which case the transition function Pt,t+h(i,j) := IP(X(t) = jIX(t) = i), i E E, j E E and t > h > 0 is independent of the time t. Thus we can put Pt,t+h(i,j)= Ph(i,j). The C h a p m a n K o l m o g o r o v equation in this case can be written as follows:

12


Pt+h(i,j) = Z P t ( i , k ) P h ( k , j ) ,

(26)

kEE

which in matrix form can be written as

Pt+h - PtPh •

(27)

Thus the family (Pt, t _> 0) forms a semi-group. We will study here Markov processes w/th semi-group satisfying the following property:

limp,(i,j) = l{i_j} t+0

(28)

for all i,j E E and the semi-group will be called standard. If limtloPt(i,i) = 1 is satisfied uniformly with respect to i E E, then the semi-group will be called uniform.

3.2. Reliability in continuous t/me Let us follow the same formulation of the reliability-related indicators as in the case of DTMC. The difference is that the t/me here is continuous. Consider a CTMC X, w/th state space E = ( 1 , . . , s}, generator A, transit/on function Pt and initial distribution 0{. As in the case of DTMC, consider the up states set U = { 1 , . . , r } , and the down states set D = {m + 1 , . . , s } and the part/tion of the generator and the initial distribution vector following U and D: A=

(An

A12)

\A21

A22

0{ = (0{1

0{2) .

Availability: PROPOSITION 3.1. The (pointwise) availability of a C T M system is g/ven by

A(t) = 0{etAls,r .

(29)

The steady-state availability, denoted by Aoo, is given by Aoo : Z ~ ( k )

: 7c-ls,r •

kEU

Reliability: PROPOSITION 3.2. The system's reliability is R(t) = 0{~etanlr.

Maintainability: PROPOSITION 3.3. The maintainability is g/ven by M(t) = 1 - 0{2etA221s_r.

Hitting times: For the mean hitting times that were defined in the previous section we have the following:


13

PROPOSITION 3.4. MTTF = - CqAllllr, MTTR

= -

Œ2A~lls r •

PROPOSmON 3.5. I f the C T M C X is ergodic, then 7"Cll m

MUT - - -

7ZlA21 l r '

7"C21N_m

MDT -

7z2A121s-r

and M T B F = M U T

+ MDT.

PROPOSITION 3 . 6 . Var(T) = 2«lAi-~l - ( C q A l ? l ) 2 .

3.3. Distributions o f phase type Phase-type distributions (of Ph-distributions) play a very important role in reliability theory for two reasons. The first one is because of a property making this family to be a dense set in the set of all distributions on IR+, while the second one is due to the fact that we have an explicit formulation of the basic system operation in reliability as parallel system, series system, redundant cold standby system, etc. (see Neuts, 1981; Asmussen, 1987). Consider a M a r k o v process, X say, with generator A and stare space E = {1,..,N+ 1} with the following partition: U = { 1 , . . , N } and D = { N + 1}. Suppose that U is a transient class and state N + 1 is an absorbing state. Following U and D the generator and the initial distribution vector can be partitioned as follows:

A:(~ A0 0) and (c~,~ZN+I). DEFIMTION 3.2. A distribution function x H F(x) on [0, oc) is o f phase type ( P H distribution), if it is the distribution function o f the absorbing time o f a C T M C defined as above. The couple (Œ, T) is called a representation o f F. The distribution of absorbing time in the state F(x) = 1 - c~exrl . Properties (Neuts, 1981):

N + 1

is given, for x _> 0 , by


14

1. It possesses an atom at x = 0, which is equal to C~N+I. The absolutely continuous part has density f , given by f(x) = F'(x) = aeXTTo, x > O. 2. The Laplace-Stieltjes transform/~ of F is given by /W(S) = ~ N + I -~- O~(SI --

T)-IT0,

for Re(s) _> 0

.

(30)

3. The moments (no centered) are ]~!. z (-1)~n!(~T-'l), n > O. PROPOSITION 3.7 (Convolution of two distributions of phase type - N e u t s , 1981).

Consider two distributions of phase type, F and G say, with representations (c~,T) and (fi, S) of order M and N, then their convolution F * G is a phase-type distribution with representation (7, L), where

0:0) and 7 = (e, CgN+lfl)" Let us define now some elements of Kronecker's algebra useful for the next proposition. The operation ® is the Kronecker's sum of two matrices. Note d//m, the space of matrices of dimension m x n and let A c J g , , and B E Jmm; the Kronecker's sum is defined as follows:

A

= (A ®Ira) + (B®I,)

with ® the Kronecker's product. Let A E ,/¢lkt and B E ~/~mn, we have: A @ B E J~kxmflxn and

allB

...

a11B)

A®B= ak~B ...

aklB

PROPOSITION 3.8 (Formation of series and parallel systems - Neuts, 1981).

Consider two phase-type distributions F and G with representations (cq T) and (fl, S) of order M and N, respectively, then: 1. The distribution K given by K(x) = F(x)G(x) is a Ph-distribution of representation (7, L) of order MN + M + N with L = (T@So0 I®SOTo T O o 1 )

and 7 = [c~@ fi, fiN+lCq C~M+lg]

2. The distribution W defined by W(x) = 1 - ( 1 - F ( x ) ) ( 1 - G ( x ) ) is a Ph-distribution of representation (c~® ~, T @ S).


15

P~OPOSITION 3.9 (Asymptotic behavior - Neuts, 1981). Let F be a Ph-distribution of representation (cq T). I f T is irreducible, then F is asymptotically exponential, i.e.,

1 - F(x) = Ke -z~ + o(e -~=) , K > O, 2 > 0 with - 2 the eigenvalue of T having the greatest modulus of real part and K = c~v where v is the right eigenvector of T corresponding to the eigenvalue

4. Semi-Markov processes and reliability Some systems satisfy a M a r k o v property not for all points of time but only for a special family of increasing stopping times. These times are the state change times of the considered stochastic process.

4.1. Basic results and definition o f a Markov renewal process 4.1.1. Definition Consider an at most countable set, E say, a two-dimensional stochastic process (J,S) = (J~,S~, n E N), where the r.v. J~ take values in E. Consider also that the r.v. Sn takes values in IP,+ and satisfies 0 = So _< $1 _< $2 _< ... DEFINITION 4.1. The stochastic process (J, S) is called a Markov Renewal Process ( M R P ) if it satisfies the following relation: lP(J~+l = j, Sn+l 0), i.e., for each time t _> 0 the r.v. N(t) is

N(t) := sup{n : Sn 0 and maxi,j SUpxI(I - H(x))(i,j)l _< 1, the solution of the above M R E exists, is unique and it is given by P(t) = (I - Q(t)) (-1) * (I - H(t)) ,

B a s i c p r o b a b i l i s t i c m o d e h ' in reliability

17

where ~9(t) = (I - Q(t)) (~) = Z Q ( " ) ( t ) n>O

is the M a r k o v R e n e w a l Function. Moreover, O!~)(t) = x-~U

Qik (t-u)Qkjdu, {0~k f ó~/~1/

ift>O ifto}, Qij(t)

.

4.1.4. L i m i t distributions and theorems

Here we give some limit theorems useful for the reliability analysis. More importantly, these theorems give an extension of the basic classical limit theorems in probability theory to the semi-Markov setting. THEOREM 4.1 (Steady-state distribution, Taga, 1963). 7rj = l i m Pij(t) = vjmj vm where

]Ei[S1]

v = (vi) :

is

aH

invariant

measur«

of

(Jn)

and m = (mi)

with

mi =

f~(1 - Hi(t))dt.

The following two theorems are straightforward applications of the Blackwell renewal theorem and of the key renewal theorem, respectively. THEOREM 4.2 (Blackwell type theorem, ~inlar, 1969). c

~,ü(t) - ~,,,(t-

«) ~

E,[sl] '

ast--+

(?~ .

THEOREM 4.3 (Key renewal type theorem, ~inlar, 1969). I f i persistent state and hi a direct R i e m a n n integrable function, then t

/o ~~~ldyl~~l~-~l ~ ~

1

oo

/o ~~~ld~

THEOREM 4.4 (Law of large numbers, Taga, 1963). 1

n

~~X~~~~[Xl],=

as t ~ o~ .

is non-periodic


18

THEOREM 4.5 (Central limit theorem, Taga, 1963).

Xl + . . . + X, - hIE[X1] --+ N(O, 1) . nv/h~io-i

Under the hypotheses that E~[X1] < oc, and a2i = IEi([Y[ - SlIEr(X1)] 2) < ec where yi = 2j=s~ 1+1Xj, S~ = 0, n = 1 , 2 , . . . , then S i, is the recurrence time of state i for the Markov chain (J~). 4.1.5. A central limit theorem Let f be a real measurable function defined on E x E x IR. Define, for each t _> 0, the functional Wf(t) by N~j(t)

Wf(t)= Z

E f ( i , j , Xij~)

i,j n=l

when the series converges. Put:

/0 ~0

Aij=

f(i,j,x)dQij(x),

Ai=

Aij, j=l

0(3

Bij =

S

(f(i,j,x))ZdQij(x),

Bi = Z B i j j=l

and s

mi =

=

ii 5 ZAJ/ j=l

-

+

Br

i;/

)j

r--1 s

+ 2E Z

E AreAklti*i(#*ti + #i*k- I~e*k)/(#~r#*kk),

r=l g,~-i k~-i

mi B f - -_ - -

mf~--~ #ii

~2 z

.

~lii

THEOREM 4.6 (Pyke and Schaufele, 1964). Under the hypotheses that the above

moments are finite, we have that, as t -+ oc, t-1/Z[wf(t)- t.mf]

d N(O, Bf) .


19

4.1.6. A functional central limit theorem

Let W(t)

=

g(Z(u))du

be a functional of the semi-Markov process (Z(t), t >_ 0). Hypothesis (H): • MC (Jn) has a unique invariant measure ~;

• llp_ 1 such that k 1

k

ZPi Var(~[XrY]) .

(38)

However, since lE{lE[X]Y]} = lE[X] = 0, lE[X]Y] can also be used as an unbiased estimator of 0 and it is preferable to do so since its variance will be smaller than the one of the direct estimator as indicated by (38). In fact in this case, we prefer to simulate the random variable Y and by observing its values make conclusions about X. In this, we suppose that lE[X] Y] is known or can be easily determined from the simulation tun. This method may be quite efficient in the case where we are interested in estimating the failure probability of a k-out-of-n system. The variable Y will represent the n - 1 components and if we know the state of these components we will probably be in position to determine the state of the system. 6.2.2. Stratified sampIing Stratified sampling is a variance reduction method based on the same formula (37) as the conditioning method. The idea behind this method is to separate our original sample space into different strata and carry out independent simulations


29

in each of these• Then, by taking the expected value of the output of the simulations on all of the strata we can obtain the desired estimate of the system's parameter. However, the principal difference with the method of conditioning is the fact that the former uses (38) to prove the variance reduction while in the case of stratified sampling the basic argument is that Var[X] _> IE(Var[XIYI). Moreover, it is sometimes difficult to separate the sample space into strata and to define how many samples we have to take from each one in order to have the largest possible variance reduction. This depends clearly on the problem at hand and may be difficult to do in the general case, except in problems having an intrinsic layered structure. See Ross (1990), for the details and examples of this method.

6.3. Importance sampling 6.3.1. Relative error Consider now the problem of estimation of the following quantity: 7 = ~y[h(x)] =

h(x)f(x)d~,

(39)

O0

where X is a real random variable defined on a probability space (~2,~ , IP), having density f and h:lR--+ IR. In order to estimate ~ using Monte Carlo simulation we can generate an n-sample (~ol,..., con) issued from f and consider the following unbiased estimator:

n

•

In the case where h(x) = lA (x) with lA (x) = 1, if x ~ A and 0 otherwise, then 7 = IP(X ~ A). Moreover, the variance of the ~n estimator is equal to 7(1 - 7)/n, while the associated 100 x (1 - 6)% confidence interval will be

where z6/2 is defined by the equation 6/2 = P(Z > z6/2) and Z denotes a random variable having the standard normal distribution N(0, 1). If we are interested in constructing a confidence interval for 7 the natural way will be to continue the simulation until the interval's half width becomes less than ~c (~c E]0, 1D times the value of the parameter that we are trying to estimate. Thus, the stopping criterion for our simulation will be

z~/2

)~(1 - ~~) < ~~, n

which implies that z~/2

7

< ~c .

(40)

N. Balakrishnan, N. Limniosand C. Papadopoulos

30

Note however, that the relative error (RE) of the estimator ~ù, which is defined to be the ratio of its standard deviation to its expected value, will be given by (as n ~ +oc)

Bl~-~ù) REG)

= z6/~

-

-

~,

,'~ Z 6 / 2

1 ~,

^ n-,+o~ since ?.

"7 •

It is this last equation that clearly illustrates the inconvenience of using direct simulation: the relative error of the estimator remains without bounds, while the event becomes rarer and rarer (i.e. RE(~n) ~ +ec when 7 --+ 0). It also means that in order for Eq. (40) to be satisfied and thus obtain the desired relative precision of estimation, we have to considerably increase the size n of the sample. In other words, in order to estimate ? up to a certain level of precision, one has to increase the number of simulation runs as the probability of the event becomes smaller and smaller.

6.3.2. Some background theory Importance sampling is a method that may help us to overcome the previous difficulty. The basic idea of the method is to change the original probabilistic dynamics of the system, and modify at the same time the function to be integrated. This change of measure is illustrated as follows: =

f

+o~ h(x ) ~ f

j ~x)

' ( ~ ) & = ~s, Eh(x)L(x)]

,

(41)

where L(X) = f ( X ) / f ' ( X ) represents the corresponding likelihood ratio and the subscript f ' means that the expected value is now taken with respect to the new density fl. The name given to this method is due to the fact that the process is sampled in the areas that are more important for the estimation of 7, in the case where h(x) = lA(x) the areas where the event {X E A} is realized. Consequently, the new density f~ has to be chosen in a way to make the rare event under consideration more likely to occur. Since this change of measure introduces a bias to our estimation, the results obtained by the simulation have to be multiplied by the appropriate likelihood ratio. This term plays the role of the compensatory factor, since the system has been simulated using a probability measure that is not directly associated to the system's model. Eq. (41) is valid only in the case that f ( x ) > 0, for every x E IP, with f ( x ) > 0 and h(x) > 0, which implies that a possible value o f X under f , is also possible under S - It is possible however to have f ( x ) = 0 and f(x) > 0, for any x E IP, with h(x) = 0. By making this change of measure, the new unbiased estimator of 7 will be ~n(f') = ~ ~ L 1 h(oi)L(°»i), where the new n-sample (col,..., con) has now been generated using density S . Its corresponding variance is given by


Varf,[~n(f')] =

~ h(x)

= IEf[h(X)L(X)] - 72

31

S ( x ) d x _ ?2 (42)

The main aim of importance sampling is to find a suitable - and easily imp l e m e n t a b l e - new density f l in order to minimize the variance of 7n(ff) and by doing this, reduce the cost of the estimation procedure. Thus, using importance sampling the rare event has to be realized more often, meaning that its new probability taust be greater than the original one. The corresponding L term in Eq. (42) has to be kept as small as possible. In the case where the L term is uniformly less than one, then Varf,[~ù(f')l < 7 - 7 2 = Varf[Tn(f)] and we will certainly obtain a variance reduction. Another alternative would be to choose f~ in a way that lEf Ih(X)L(X)] is of the same order of magnitude as 72. In such cases the associated change of measure is sometimes called asymptotically efficient or asymptotically optimal (see the survey of Heidelberger, 1995 for a discussion on this matter). An optimal change of measure is defined to be a measure that results in a zero variance estimator for the unknown quantity (see Kuruganti and Strickland, 1995) and it always exists. For our example, this corresponds to choosing

f* (x) -- h(x)f(x) ?

(43)

Using the optimal change of measure for the simulation, the exact value of the parameter will be obtained in the first simulation tun. Unfortunately, it has the disadvantage of containing 7, the parameter that we are trying to estimate, making it thus not directly exploitable. Nevertheless, in some special cases, we can explicitly construct this optimal change of measure, which will enable us not only to estimate ? at a minimum cost, but also - and more importantly - to find its exact value, as a by-product of the intermediate calculations (see Kuruganti and Strickland, 1995, 1997). The conditions on the applicability as well as the theoretical framework behind importance sampling are given in Glynn and Iglehart (1989). In their work, importance sampling is extended to problems arising in the simulation of both discrete time and continuous time Markov processes, as well as in generalized semi-Markov processes. In the same paper, the authors discuss the problem of steady-state quantities estimation, that can be carried out by exploiting the regenerative structure of the Markov chain, as well as the estimation of transient quantities, where a different approach has to be used.

6.3.3. The optimal change of measure Let us consider again expression (43) and let h(x) = lA (x). In this case, the choice f* (x) = f ( x ) 1~ (x) 7

(44)

32

N. Balakr&hnan, N. Limnios and C. Papadopoulos

corresponds to the optimal change of measure associated to the estimation of 7 = IP(X E A). Even though Eq. (44) seems to be at first sight of no use, since it contains the unknown parameter, it has the benefit of providing us with a very useful insight concerning the choice of the new density fl(.). Indeed, as it is indicated in Strickland (1993) the following hold: • All the mass of the probability is concentrated on the rare event {X E A}, and consequently only those samples that correspond to the realization of this event will be produced when the optimal change of measure is used to carry out the simulation. • On A, the new density is exactly the conditional density of X, given that {X C A} has occurred: f ( X ) " IA(x) _ ~ f ( x [ X E A), f * (x) -- ~ ~- Ä) 1, o,

x E A, otherwise .

Therefore, the relative likelihood of the values of X on A, is exactly the same for the original as well as the new distribution dF*(xl) _ dF(xl) dF* (x2) dF(x2) '

where Xl,X 2 ŒA

and F(.) represents the cdf corresponding to the density f ( . ) .

6.4. M a r ovian systems and importance sampling Consider now a continuous time M a r k o v chain X = {Xt: t > 0}, with state space E = {0, 1 , . . , s } , s < +ec, infinitesimal (conservative) generator Q = {q(x,y): x , y c E}, and initial law #(.). The quantity q(x) = - q ( x , x ) represents the total rate out of state x. Suppose also that the stare space of the system is divided into two disjoint subsets, U and F, with U U F = E and U N F = (3. The set U = {0, 1 , . . , m } represents the set of operational states, while F = {m + l , . . . , s} stands for the set of failed states of the system. In state 0, all components are considered new. Define also 0 = To < T1 < ... < Tn < ..., the sequence of the successive j u m p times of the chain X. Then Y = {Yn, n > 0}, defined by B-Xtù,

n=0,1,..

,

will be the embedded discrete time M a r k o v chain associated to Xt. The elements ofits transition matrix P = { P ( x , y ) : x , y E E}, are given by P(x,y) = q ( x , y ) / q ( x ) , when x ¢ y and 0 otherwise. Importance sampling can be easily extended to the case of Markovian systems. In order to modify the probabilistic dynamics of the system, one has to basically modify the transition probability matrix or the generator of the process (see Glynn and Iglehart, 1989) and/or the initial distribution of the process.


33

The estimation problems may concern transient or steady-state quantities. In the first case, the regenerative structure of the chain is employed and the estimation problem is transformed to its analog over the regenerative cycles of the chain (see the following discussion), while in the second case the process is simulated for a given time horizon. Moreover, in case of simulation of a continuous time Markov chain, one part of the simulation is devoted to the simulation of the corresponding embedded Markov chain where the sequence of states visited by the process is generated. Then, given the sequence of states visited by the chain, the second part of the simulation concerns the generation of the associated (conditional) sojourn times in these states. This discrete time conversion always results in a variance reduction (see Goyal et al., 1992; Fox and Glynn, 1986).

6.4.1. Regenerative simulation Let -c be a stopping time for {Yn : n _> 0} which means that the realization or not of the event {z = n} may be determined by yn _ (Y0,. •, Yn). Note also En for the set of all possible paths of the chain Y until time n en - { / =

( y 0 , y , , . . ,yn): • c E} .

Then, the probability associated to any yn c En is given by P(Y") = #(Yo)P(Yo,Yl)...P(Yn-I,Yn) , where #(Y0) = P(Y0 = y0) is the initial law of the chain. Moreover, let Bn c f~n stand for the set of all paths for which {z = n}. We have the following proposition: PROPOSmON 6.1 (Goyal et al., 1992). Consider a discrete time Markov chain with transition probability matrix P. Ler P be the probability measure associated with the different trajectories of the chain and z a stopping time which is finite under P, with probability 1. Note also Z, a measurable function of Y~ .for which IEe [[Z(Y~)11 < ~ . Let P' be a new probability measure for whieh z is also finite with probability 1 and for any y~ ¢ Bù, p'(yn) • 0 whenever Z(yn)P(y n) ~ O. Then IEe[Z(Y~)] = IEp,[Z(Y ~) L(Y~)], with L(Y ~) = P ( S ) / P ' ( y n ) , for any yn C Bh. Remark however, that in this case it is not necessary for the new importance sampling measure to correspond to a time-homogeneous Markov chain. A different measure P' given by p,(yn) = P'(Yo)P'(Yl lYO)'" P'(Yn]Y0...Yn-1) , can also be used and it is called a "Dynamic Importance Sampling measure" (DIS, see Shahabuddin and Nakayama, 1993 and references therein). In this, P'(YnlY0..-Yn-1) represents the likelihood of the path Yn=yn given that En-1 = CVO,... ,Yn 1).

Consider now that we are interested in estimating the steady-state unavailability of the system c~that represents the fraction of time for which the system is


34

considered failed. Let h(y) = 1/q(y) be the mean sojourn time in state y and let g(Y) = 1F(y)h(y). Then, we can write (see Crane and Iglehart, 1975) nz rv,~0-1

« = ~PL~k=0

g(:~k)]

(45)

] l P rV'VCo -1 h(rk)l L2Jk=O

Let us now define TB = inf{t > 0 : Xt E B}, the hitting time for B C E (with the convention that infO = +co) and r» = inf{n > 0 : I1, E B}, the number of jumps for Y to enter B C E. A somehow similar representation holds for the mean time to failure (MTTF) of the system, which may be written as (see Goyal et al., 1992) nz rv'~min(z0,ZF) - I h(Yk)] M T T F = ]Ep[TF] = ]te[min(T0, TF)] = uzP[2--*k=0 P(TF < To) ]lp[l{zF 0,

exp - fö I-H~(ù)J

0

(49)

otherwise .

The semi-Markov matrix Qij(t) is given by Qij(t) =

hi«(u)Aij(du),

i,j E E ,

where 1 - Hi(u)

hij(u) - 1 - Aij(u) -- IE[~j]z/j = u 1 and Iij is the indicator function o f the event {mink~Æ zik = zij}. The algorithm for the realization of one sample trajectory of the process is the following.

Algorithm 4. I n p u t data: The state space of the process, its initial law #(.), and the distribution functions Aij. 1. 2. 3. 4.

Sample a r.v. X ~ # and set t = 0, X(t, o~) = X(m); Set i = X(t, ~o); generate zifs (/" C E) using the distribution functions Aij; Set ~ = minj~s ~,ij, t = t + z, set X(t, o~) = arg minjcs Tij'~ Repeat steps 2-3 for the n u m b e r of j u m p s of the process needed, or until the time t becomes greater than the observation period T.

7.2. Embedded M a r k o v chain method This m e t h o d consists simply in using the transition probabilities of the e m b e d d e d M a r k o v chain in order to find the next state of the system and generating the Exp(0.001)

~x~~oo~~~®/we~~u~~~o~~~ Fig. 4. A 3 states semi-Markov system.


39

Availability for the semi-Markov system ! ! !

1

\

--......

0.99

. . . . . . . !. . . . . . .

!

!

Competin Risk Embedde~ Markov Chain Method

l --

Analytical solution

]

0.97

0.94[.......

i.........

i

!

.................................................

|

0.920"93I. ...... . . . . . .i. . . . . . . . . . . !.

0.91

.......

0

50

100

1 ~

!

!

i

i i

150

" >'~"~ . . . . . . .

i

200

250 Time

! .......

300

i. . . . . . . . .

........

350

400

450

Reliability for the semi-Markov system ~ ! ! ~

~

!

500

Competin q Risk Embedded Markov Chain Method

....

0.950.850.9

I

I

....................

0.6 0.75 07

0.65

0.6

50

100

150

200

250 Time

300

350

400

Fig. 5. Graphical comparison of the simulation methods.

450

500

40


corresponding holding times to the states visited using the distribution functions F/j(t). This algorithm is similar to the algorithm used for the simulation of C T M C and is given below.

Algorithm 5. Input data: The state space of the process, its initial law /~(.), the transition probabilities Pij and the distribution functions ~j. 1. Sample a r.v. X ~ # and set t = 0, X(t, co) = X(co). 2. Set i = X(t, co); using the transition matrix of the embedded M a r k o v chain find the state j in which the process will jump into. 3. Generate z, the holding time in state i, using the distribution function F~j(t),

4. S e t t = t + ~ , X ( t ,

co)=j.

5. Repeat steps 2M for the number of jumps of the process needed, or until the time t becomes greater than the observation period T. EXAMPLE. Consider the semi-Markov system whose state diagram is given in Figure 4. The state space of this system is E = {0, 1,2}, with U = {0, 1} and F = {2}. The different laws governing the sojourn time of the process in different states are indicated in the figure. Thus the sojourn law of the system in state 0 is exponential with parameter 21 -- 0.001. When the system is in state 1 and jumps towards state 0, then the sojourn time has a Weibull distribution with parameters cq = 2.0, fil = 10.0, while when the system goes towards state 2, the sojourn time has a Weibull distribution with parameters «2 = 0.7,/~2 = 2.0. When the system is in state 2, only one transition is possible, the one going to stare 0. In this case, the sojourn time in state 2 has an exponential law with parameter 22 = 0.01. We have simulated the system described previously in order to estimate its availability A(t) = IP(X(t) E U), as well as its reliability R(t) = ]P(TF > t). The results of the simulation obtained for 100 iterations of the algorithm are given in Figure 5, where the results obtained by the previous two simulation methods are compared to the results obtained using an analytical method. For the embedded M a r k o v chain the transition probabilities were taken to be pl0 = 0.1048 and P12 = 0.8952. Note however that the time step of the algorithm was 0.2 time units.

Hitting times: Mean Mean Mean Mean

time to failure: M T T F = 1119.7, time to repair: M T T R = 100, up time: M U T = 1117.1, down time: M D T = 100.

References Asmussen, S. (1987). Applied Probability and Queues. Wiley, New York. Bratley, P., B. L. Fox and L. E. Schrage (1987). A Guide to Simulation. Springer, Berlin.


41

~inlar, E. (1969). Markov renewal theory. Adv. Appl. Probab. 1, 123 187. Crane, M. A. and D. L. Iglehart (1975). Simulating stable stochastic systems III, regenerative processes and discrete event simulation. Oper. Res. 23, 3345. Fishman, G. S. (1996). Monte Carlo. Concepts, Algorithms and Applications. Springer Series in Operations Research, Springer, New York. Fox, B. L. and P. W. Glynn (1986). Discrete time conversion for simulating semi-Markov processes. Oper. Res. Lett. 5, 191 196. Glynn, P. W. and D. L. Iglehart (1989). Importance sampling for stochastic simulations. Manage. Sci. 35, 1367 1392. Goyal, A., P. Heidelberger and P. Shahabuddin (1987). Measure specific dynamic importance sampling for availability simulations. In 1987 Winter Simulation Conference Proceedings, pp. 351-357. IEEE Press. Goyal, A., P. Shahabuddin, P. Heidelberger, V. F. Nicola and P. W. Glynn (1992). A unified framework for simulating Markovian models of highly reliable systems. IEEE Trans. Comput. 41(1), 36-51. Goyal, A., S. S. Lavenberg and K. S. Trivedi (1987). Probabilistic modeling of computer system availability. Ann. Oper. Res. 8, 285 306. Hammersley, J. M. and D. C. Handscomb (1964). Monte Carlo Methods. London, Methuen. Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. A C M Trans. Modeling Comput. Simul. 43-85. Ionescu, D. C. and N. Limnios, (Eds.) (1999). Statistical and Probabilistic Models in Reliability. Birkhäuser, Boston. Janssen, J. and N. Limnios, (Ed.) (1999). Semi-Markov Models and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands. Kemeny, J. G. and J. L. Snell (1976). Finite Markov Chains. Springer, Berlin. Korolyuk, V. S. and A. F. Turbin (1982). Markov Renewal Processes in Problems of Systems Reliability. Naukova Dumka, Kiev (in Russian). Kuruganti, I. and S. G. Strickland (1995). Optimal importance sampling for Markovian systems. In Proceedings of the 1995 IEEE Systems, Man and Cybernetics Conference. Kuruganti, I. and S. G. Strickland (1997). Optimal importance sampling for Markovian systems with applications to tandem queues. Math. Comput. Simul. 44(1), 61 80. Limnios, N. (1996). Dependability analysis of semi-Markov systems. Reliab. Eng. and Syst. Safety. Limnios, N. and G. Oprisan (1997a). A general framework for reliability and performability analysis of semi-Markov systems. In Eighth International Conference on ASMDA. Anacapri (Napoli). Italy, June 1997. Limnios, N. and G. Oprisan (1997b). A general framework for reliability and performability analysis of semi-Markov systems. Appl. Stochast. Models Data AnaL (to appear). Limnios, N. and G. Oprisan (1997c). Semi-Markov process to regard of their application. World Energy Syst. J. 1(1), 6zF75. Limnios, N. and G. Oprisan (1999a). Invariance principle for an additive functional of a semi-Markov process. Rer. Roumaine. Math. Pures Appl. 44(1), 75-83. Limnios, N. and G. Oprisan (1999b). Semi-Markov Processes and Reliability. Birkhäuser (to appear). Nakayama, M. K. (t994). A Characterization of the simple failure biasing method for simulations of highly reliable Markovian systems. A C M Trans. Modeling Comput. Simul. 4(1), 52-88. Nakayama, M. K. (t995). Asymptotics for likelihood ratio derivative estimators in simulations of highly reliable Markovian systems. Manag. Sci. 41, 52zP554. Nakayama, M. K. (1996). General conditions for bounded relative error in simulations of highly reliable Markovian systems. Adv. Appl. Prob. 28. Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore, MD. Opri~an, G. (1999). On the failure rate. In Statistical and Probabilistic Models in Reliability (Eds. Ionescu and Limnios).

42


Ouhbi, B. and N. Limnios (1996). Non-parametric estimation for semi-Markov kernels with application to reliability analysis. Appl. Stoch. Models Data Anal. 12, 209-220. Ouhbi, B. and N. Limnios (1997). Estimation of kernels, Availability and Reliability functions of semi-Markov Systems, In Statistical and Probabilistic Models in Reliability (Eds. Ionescu and Limnios). Platis A., N. Limnios and M. Le Du (1998). Hitting time in a finite non-homogeneous Markov chain with applications. Applied Stoch. Models Data Anal. 14, 241-253. Pyke, R. (1961a). Markov renewal processes: definitions and preliminary properties. Ann. Math. Stat. 32, 1231-1242. Pyke, R. (1961b). Markov renewal processes with finitely many states. Ann. Math. Stat. 32, 1243-1259. Pyke, R. and R. Schaufele (1964). Limit theorems for Markov renewal processes. Ann. Math. Stat. 35, 1746-1764. Ripley, B. D. (1987). Stochastic Simulation. Wiley, New York. Ross, S. M. (1990). A Course in Simulation. Maxwelt MacMillan International Editions. Shahabuddin, P. (1994). Importance sampling for the simulation of highly reliable Markovian systems. Manag. Sci. 40, 333-352. Shahabuddin, P. and M. K. Nakayama (1993). Estimation of reliability and its derivatives for large time horizons in Markovian systems. In 1993 Winter Simulation Conference Proceedings, pp. 422429. IEEE Press. Strickland, S. G. (1993). Optimal importance sampling for quick simulation of highly reliable Markovian systems. In 1993 Winter Simulation Conference Proceedings, pp. 437~444. IEEE Press. Taga, Y. (1963). On the limiting distributions in Markov renewal processes with finitely many states. Ann. Inst. Star. Math. 15, 1-10.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

") /_...,

The Weibull Nonhomogeneous Poisson Process

Asit P. Basu and Steven E. Rigdon

The Weibull h a z a r d function is often parametrized as

h(x) = u

(x_k ~-1 \0/

,

x > 0

(1)

x > 0 .

(2)

or

h(x)

= ,~fix/~-x,

The n o n h o m o g e n e o u s Poisson process that has an intensity function o f f o r m (1) or (2) is often called the WeibuU proeess, or m o r e c o m m o n l y , the power law proeess. Such a process is often used to model the occurrence o f events in time, and in particular, to model the failure times of repairable systems. We therefore begin with a discussion of the h o m o g e n e o u s and n o n h o m o g e n e o u s Poisson processes.

1. The Poisson processes Let N(t) denote the n u m b e r of events that occur at or before time t. Such a r a n d o m variable is called a eounting proeess. W h e n the argument to N is an interval, such as (a,b], then N(a,b] is defined to be the n u m b e r o f events that occur in that interval. Thus, N(t) = N(0, tl. A counting process N(t) is said to be a Poisson process if: 1. N(0) = 0. 2. F o r any a < b _< c < d the r a n d o m variables N(a, b] and N(c, d] are independent. This property is called the independent inerements property. 3. There is a function 2, called the intensity fnnetion, such that

2(t)

= lim At-+O

P(N(t, t + At] =

1)

At

4. lim At~O

P(N(t,t+ At] > 2) At

= 0 .

The last property precludes the possibility o f simultaneous failures. 43

44

A. P. Basu and S. E. Rigdon

These four properties, as minimal as they seem, are enough to establish the property that the number of failures in the interval (a, b] has a Poisson distribution with mean equal to

Ja b E(N(a, b]) =

2(t)dt .

The proof involves recursively solving a system of differential equations; see Rigdon and Basu (2000) for the derivation. The function

A(t) =

~0t ,Z(x)dx,

which gives the expected number of events through time t, is called the mean funetion for the process. Clearly, A'(t) = 2(t). The nonhomogeneous Poisson process having an intensity function of the form

=0~,0/

,

t>0

(3)

or ,~(t) = ,t/~t ~-~,

t> o

is called the power law process or the Weibull nonhomogeneous Poisson process. This model has gone by many other names as well, including and most notably, the Weibull process. When fi < 1 the intensity is a decreasing function of t. In this case, failures will become less frequent as the system ages; this is reliability improvement. When fl > 1 the intensity is an increasing function of t, and in this case failures will become more frequent as the system ages. This is called deterioration. When fi = 1, then the intensity function is a constant. Thus, the homogeneous Poisson process is a special case of the power law process. The power law process can thereYore be used to model systems that improve, deteriorate, or remain steady over time, hut it cannot be used to model systems that improve for some intervals of t and deteriorate for other intervals. Some repairable systems have an intensity function that has the bathtub shape as shown in Figure 1. For small values of t, that is, when the system is young, the rate of occurrence of failures (ROCOF) is high and failures are frequent. After the bugs are removed, or after some of the weakest components fail, the R O C O F will be smaller, and it will remain at this level throughout its useful life. Then as the system ages, the R O C O F begins to increase. At this stage, the system is deteriorating. The two functions in Figure 1 look nearly identical, but there is an important difference in their interpretations. The bathtub intensity function indicates that the system will initially experience reliability growth. A few early failures will be followed by the useful life when failures occur at roughly a eonstant rate.

The Weibull nonhomogeneous Poisson process ~(t)

Bathtub intensity function

h(x)

45

Bathtub hazard function

0.390 we would reject the null hypothesis that the power law process is an adequate model. Since the percentage point corresponding to = 0.001 is 0.512, the P-value for this test is less than 0.001. Rigdon (1989) suggested a second transformation that leads to a goodnessof-fit test for the power law process. The log-ratio transformation is Ui=log(tn/&-i),

i= l,2,

..,n-1

.

If the null hypothesis of the power law process is true, then given T, = &, the random variables Tl, T 2 , . . , Tn_l are distributed as n - 1 order statistics from the distribution with cdf

{

oB/ y < 0, G(y)= th)~, 0_ 1) and deterioration (fl < 1), Rigdon and Basu (1988, 1989, 1990) suggested using fl = 1 in the expressions above, yielding the estimators 2UB=(n--1)(n--2)fi ntn

60


and BMMSE = (n -- 2)(n -- 3)]~ ntù The first estimator is unbiased when fi = 1, and the second has m i n i m u m m e a n squared error within the class {c~ß/tn} when fi = 1. The estimator )~MMSE is a good estimator, in the sense of having a small MSE, for a wide range of il's. F o r the time truncated case, Rigdon and Basu (1990) studied the estimators BUB = (N - 1)(N - 2)fl Nt and ,~MMSE = (N - 2)(N - 3)fl Nt The estimator )~MMSE had a high efficiency relative to the M L E for a wide range of il's. Rigdon and Basu (2000) give tables for calculating confidence intervals for 2(tù) (failure truncated case) or 2(t) (time truncated case). These tables give values col and co2 (failure truncated case) which yield the confidence interval

o~l~(tù) < ,z(to) < ~o2~(tù) and Pl and P2 (time truncated case) which yield the confidence interval

pl'(t) < x(t) < p2~(t) . EXANPLE 2 (Failures of copy machine). There were n = 18 failures for this machine, and the process was failure truncated. We found previously that B ~ 1.08055 and tù---196877. The M L E ,~(190877) and the M M S E estimator 2MMSE(196877) are )~(196877) - 18 x 1.08055 ~ 0.000099 196877 and )~MMSE(196877) = (18 -- 2)(18 -- 3)1.08055 18 X 196877 ~ 0.000073 . Units for these estimates are "failures per copy".

7. Multiple systems modeled with the power law process So far, we have assumed that we observe a single system. In practice, however, we often observe a n u m b e r of repairable systems. F o r example, a rental car c o m p a n y

The Weibull nonhomogeneous Poisson process

61

may keep repair records on each car in its fleet. In a case like this, we observe multiple failures on multiple systems. H o w we handle the data analysis for multiple systems depends on what assumptions we are willing to make regarding the systems. Consider the following: 1. All systems are identical, in the sense that the failure process for each system is the power law process with parameters/~ and 0. (/3 and 0 are the same for all systems.) 2. All systems are different and each is modeled by its own power law process. (The ith system is modeled by a power law process with parameters/3i and 3. All systems have the same growth parameter/3, but possibly different 0's. 4. All systems are different, but they are similar enough to assume that the parameters of the power law process are drawn from some prior distribution.

Oi.)

Under the second assumption, each process is estimated separately using the methods of the previous section. We now discuss separately the other three cases.

7.1. Identical power law processes N o w we assume that all k systems are identical and that they are modeled by a power law process with parameters 0 and /3. This is a strong assumption and should not be made without justification. Let denote the time of the j t h failure on the/th system. Also, let ni denote the number of observed failures for system i, and N = nl + n2 + • • • + nk. We assume that system i is observed until time T~, where T, = ti,~i (failure truncated case) or T~ > (time truncated case). Because the failures on separate systems are independent, the likelihood function is

tij

ti,ni

= /3N o-ô~i=In~ l~j~_ltij ) g(O,/3)

k exp

L(O,/3)

Differentiating the log-likelihood = log with respect to 0 and with respect to/3, setting the results equal to zero, and simplifying, yields

O= ( ~~=l liß ) 1/~ N

-

(19)

and B = O-fl

N ~~=1 T/Blog T,, - ~k=l Ejril log t/j

(20)

The expression for 0 in (19) can be substituted into (20) yielding one nonlinear equation in one unknown. Once the estimate/) is obtained, it can be substituted

62


into (19) to obtain the MLE of 0. For the special case when all k systems are time truncated at the same time T, the MLEs have the closed form expressions N

B

:

Ek=l ~~/1 log(T/tij)

and

ô_

kl/~T N1/~

If we condition on the time of the nth failure (failure truncated case) or on the number of observed failures (time truncated case) we obtain a conditional likelihood function. This conditional likelihood can be used to find the conditional maximum likelihood estimators (CMLEs) for ig and 0 (Crow, 1974)

ß

a4 log(Ti/@

(21)

= 2~=a 2>,

and

õ=(

~~~~)l~~

(22)

where

mi =

{

ni - 1 if data are failure truncated, ni if data are time truncated

and

k M= Zmi i=i

.

Since the estimator ]~ has the property that 2nfl/fi ,--, X2 (2M), we have

\

2M

< ig
0 for t > to. ttere the strict monotonicity is implied. For example, when h(t) is increasing, it is strictly increasing. The bathtub curves given in this definition would probably represent some U-shaped tubs rather bathtubs as there is no interval for which h(t) is a constant. If F is not absolutely continuous, we may define B F R through the conditional reliability function F(x]t)-Æ(t+x)= ,

F(t)

F(t)=l-F(t)>O

.

(3.1)

c. D. Lai, M. Xie and D. N. P. Murthy

74

DEFINITION 2 (Haupt and Scabe, 1997). F is BFR if there exists a to such that • /~(xlt) is increasing in t for 0 _< t < to, 0 _< x _< to - t, • F(xlt ) is decreasing in t for to _< t < oc,x >_ 0. Definition 1 may be modified to allow for a more 'comfortable' bathtub shape shown in Figure 1. DEFINITION 3 (Mitra and Basu, 1995). A life distribution F which is absolutely continuous and having support [0, oc) is said to be a BFR distribution if there exists a to _> 0 such that h(t) is non-increasing for [0, to) and non-decreasing on

[to, oo). In this definition, the 'flat' part of the bathtub is allowed though not explicitly. to is refereed to as a change point of the distribution F by the above-named authors. A more explicit definition that gives rise to curves having definite 'bathtub shapes' is as follows. DEFINITION 4 (Mi, 1995). A distribution F is a bathtub shape life distribution if there exists 0 _< tl _< t2 < oc such that: (a) h(t) is strictly increasing if 0 < t < tl; (b) h(t) is a constant if tl _< t _< t2; and (c) strictly increasing if t _> t2. Several comments are in order: 1. In Definition 4, Mi called the points tl and t2 as the change points of h(t). If tl = t2 = 0 , then a BFR becomes an IFR; and if h = t2 ---+oo, then h(t) is strictly decreasing so becoming a DFR. In general, if q = t2, then the interval for which h(t) is a constant degenerates to a single point. In other words, the strict monotonic failure rate distributions I F R and D F R may be treated as the special cases of BFR in this definitions. Park (1985) also used the same definition. 2. In Definition 4, at most two change points are allowed. In other words, the points in the interval (tl, t2) are not change points according to Mi (1995) but would have been called the change points according to Mitra and Basu (1995). 3. Definition 4 may be rewritten as

h(t) =

{

hl(t) 2

for t_< h, for tl _< t _< t2,

h2(t)

for t _> t2 ,

(3.2)

where hl (t) is strictly decreasing in [0, t] and h2(t) is strictly increasing for t _> t2. We do not know of many parametric distributions that possess this property. However, Jiang and Murthy (1997c) found a distribution while studying sectional models involving three Weibull distributions. More details will be given in Section 4.1.

Bathtub-shaped failure rate life distributions

75

4. We may differentiate the types of bathtub failure rates based on the asymptotic nature of h(t) as t approaches 0 and infinity, h(t) may be finite or infinite at these asymptotes. Following Mitra and Basu (1996a), we let {BFRs} denote the family of the bathtub distributions covered by Definition 1 with the subscript s standing for "strict". We use {BFR} to denote the family of bathtub distributions covered by Definitions 3 and 4. Clearly, {BFR} = {BFRs} U {IFR} U {DFR}. Another definition of a bathtub shaped failure rate distribution may be defined through logff(t). DEFINITION 5 (Deshpande and Suresh, 1990). A life distribution F having support on [0, oc) is said to be a bathtub shaped failure rate distribution if there exists a point to such that - log/~(t) is concave in [0, to) and convex in It0, oc). It is to be noted that the above definition of a BFR distribution is quite general and extends the idea of distributions possessing a bathtub shaped failure rate to situations where the failure rate itself does not exist. In studying the property of B F R class, in particular its relationship with other classes, one has to be aware which definition of BFR is used.

3.3. Some basic properties Mitra and Basu (1996a) present some basic properties concerning the bound for the survival functions and moments of a BFR distribution. Closure properties of the BFR class under the formation of coherent systems, convolutions and mixtures are also dealt with. • Su~pose F is BFR, then Æ(t) O.

• A life distributlon F in (BFR} with mean equal to e:rk - r ( ~ + 1)

(h(t0)} k is necessarily an exponential. • Convolution of distributions in {BFRs} is not necessarily in {BFRs}. In fact, even the {BFR} class is not closed under convolution. This is seen in the following example: B(t)=½(e -t+e-t/2),

t>0;

G ( t ) = e -¢ .

Then the failure rate function of the convolution H = F , G is given by 0.5, h(4) mm 0.5533 and h(t) ----+ 0.5 - - (t_l)e-t+e te-t+2e t/~2t/2. , accordingly, h(0) mm 0, h(2) Z as t --, oc which does not have a bathtub shape. • The mixture of BFR distributions need not be BFR.

h(t)

C. D. Lai, M, Xie and D. N. P. Murthy

76

• Suppose we have a competing risk model: Æ(t) =/~1 (t)f'2(t) where the lifetime of each component is BFR with a common turning point to. Then the lifetime of the system again has a BFR distribution with to as one of its turning points. (Note: the turning point here is defined as that given in Definition 3. Also a competing risk model is simply a series system). • A parallel system of two independent components from {BFR} need not be BFR. 4. Families of bathtub-shaped failure rate distributions

Many parametric families of bathtub-shaped life distributions have been constructed over the last two decades. Ideally, we should classify them into groups or strata according to some comlnon characteristics. However, this exercise seems untenable. Instead, we summarize them into two categories: (a) lifetime distributions that have explicit expressions for failure rates and (b) distributions whose failure rate functions are unwieldy or unknown. For the latter, we give only either the probability density function or the distribution function, whichever is more convenient. Behaviour of h(t) at the origin or infinity is given whenever is possible. For Sections 4.1 and 4.1.1, we endeavour to list the failure rates in the increasing order of sophistication.

4.1. Bathtub distributions with expIicit failure rate functions Quadratic and generalization. • Bain (1974, 1978), Gore et al. (1986) considered quadratic failure rate:

h(t) = o:+flt+Tt2;~ > 0, fl < 0,7 > 0 , which has a bathtub shape, h(0) = c~, h(t) ~ oc as t --+ ec. • It is easy to verify that h(t)= exp{h(t)} has a bathtub shape if h(t) has a bathtub shape. For example, BT 9 of Rajarshi and Rajarshi (1988) listed h(t) = 1 9 exp{c~ +/~t + 7t2}, c~,7 -> O, 0 > fl _> -2(c~7) /- as havlng a bathtub shape. • In fact, any increasing function of a bathtub failure rate in itself has a bathtub shape failure rate.

A competing risk model. k(t)=l+fl~t+?~t~-I ,

c~,fi,? > 0; c ~ > 2 .

(Murthy et al. 1973). h(0) - cqh(t) ~ oc as t --+ oc. • This may be considered as a competing risk model involving a Lomax distribution (the Pareto distribution of the second kind) and a Weibull distribution. • The case when c~ = 2 is considered in Hjorth (1980): c~ ~ß

h(t) = 1 +/~---5÷ 27t,

A flexible family.

7 _< 5 -

77

Bathtub-shaped faiIure rate life distributions

• Gaver and Acar (1979) have proposed a model that has a bathtub-shaped failure rate given by h(t) = ~ + 9(t) + k(t) where 9(t) > 0 is a decreasing function t such that limt-~,o 9(t) --+ O, k(t) is a decreasing function of t such that k(0) = 0 and limt~~ k(t) --+ ec and 2 is any real number such that h(t) > O. • For example, 0 h(t)=2+t~~+etP

,

e, 0_>0; t , q ) , p > O ,

which is an extension of Murthy et al. (1973) and studied by Jaisingh et al. (1987). If both 9(t) and k(t) are failure rate functions, then this model is simply a competing model involving three distributions. • Other special cases are: Canfield and Borgman (1975): h(t) = 01cqt «~-~ + 02 + 03~3/«3-1, 0~3 > 2, ~1 < 1 h(t) ---+oc as t ~ 0 or oc. Hjorth (1980): h ( t ) = c ~ / ( l + f i t ) + T t ; 0 < 7 _ < « / ? , h(0)=c~, h(t)--+oc, as t ---+oo.

Makeham's curve. • This is listed as No. 15 in Section 5 of Rajarshi and Rajarshi (1988): h(t)=3exp(#t)+e/?(l+fit) -j,

#6<e/?2,

t>0

,

h(O) = 6 + «/?, h(t) ---, oc as t ---+oc . Reliability models involving two or more Weibull distributions. Competing risk model. • Xie and Lai (1995) and Jiang and Murthy (1997c) considered a competing risk model involving two Weibull distributions resulting in a failure rate:

h(t)=/?1tll~1t/h-l+/?2q2132t#2

1 t]i>0,

/?i>0,

i=1,2.

h(t) has a bathtub shape when/71 < 1 and/?2 > 1. The turning point to given by

tO = ~/?1(1 --/?1)(t/2)/~2~I/(Ô2-Pl) (&(1--~ß2)~J

"

Also h(0) = h(oc) = oc. Sectional model with two WeibuIl distributions. • Murthy and Jiang (1997) consider two sectional models involving two Weibull distributions having failure rate functions given by

{ (/71/~1)(t/~1)~1-1, o < t < So, satisfying two conditions: to = [/?@/r/2B2]1/(/~~-&), 7 = ( 1 - / ? ) t o so that h(t) is continuous at to.

C. D. Lai, M. Xie and D. N. P. Murthy

78

• The second model is the same as above except that y = 0. Now the shift parameter y has no influence on the shape type of the failure rate. Thus the two models are essentially the same. For//1 1. • Jiang and Murthy (1997b) extended their results to give two sectional models involving three Weibull distributions. Four types of bathtub shapes are possible for both models.

Exponen tial power. • Smith and Bain (1975, 1976), Dhillon (1981), Paranjpe et al. (1985), Paranjpe and Rarjasi (1986) and Leemis (1986) studied h(t) = b//(//t)b-le (~t)~, / / > 0, b = 1/2 . h(t) --+ oc when t --+ 0 o r t --+ e « F o r / / = 1, the middle part of the bath is quite flat. B(t) has a rather simple expression, i.e., F(t) = e x p { - ( e (~t)b - 1)}. Double exponential power. h(t) =//c~t ~-1 exp(//t ~) exp[exp(//t~) - 1],c~ < 1 (Paranjpe et al. 1985; Paranjpe et al. 1986). Clearly, h ( t ) ~ t -+ ec. The survival function Æ(t) is quite straightforward,

ec as t - + 0 or

4.1.1. Finite fange distribution families Power-function distribution. • Mukherjee and Islam (1983) (see also Lai and Mukherjee, 1986) proposed a finite fange distribution with failure rate: h ( t ) - ~ptp-1 Ztp ,

OT, (1 +

O,

2p)t/T,

0 ~2, ~2 > o, e2 > 1

81

Bathtub-shapedfailure rate life distributions

1.0

/° OF(t) ///

/'/

/

!

i/ 1-v

0.(

El.0

Fig. 2. TTT-transform of a piecewiseexponentialdistribution. He claimed that this family has a bathtub-shaped but several authors contradicted his result, for example, Glaser (1980), Pamme and Kunitz (1993), and Jiang and Murthy (1998) all showed that F defined above can never be a BFR distribution. Unfortunately, several other books cited the result of Kao (1959). 4.4. S o m e comments on the bathtub shapes

Almost all the parametric distributions given above do not have an interval for which h(t) is a constant. However, many of these would have a nearly flat middle part if the parameters are chosen properly. It appears that only a sectional model can achieve a constant to describe the 'useful' life sector. It is pointed out by Haupt and Schabe (1997) that for many of these BFR distributions, estimations of parameters often resort to extensive iterative procedures. Furthermore, the main characteristics of these distributions such as moments and quantiles are not available in closed forms. For these reasons, we have not discussed the subject of estimations for BFR distributions in this review.

5. Construction techniques for BFR distributions There are many ways of constructing BFR distributions. The following may not be an exhaustive list.

82


Glaser's technique.

• Glaser (1980) chose a function ~/(t) which fulfils the following criteria: (a) ~/(t) -- - f ' ( t ) / f ( t ) and f ( t ) is a density function; (b) there exists a to > 0 such that ~/'(t) < 0 for all t c (0, to), ~/'(t0) = 0 and ~/'(t) > 0 for all t > to; (c) there exists a Y0 > 0 such that Jyõ [f(y)/f(yo)]q(yo)dy - 1 = O. Convex function.

• W i t h the definition of a BFR distribution, we can define a BFR by choosing a positive convex function h(t) over (0, oe) such that f ö h(t)dt = ce (Rajarshi and Rajarshi, 1988). Function o f random variables.

• This procedure is due to Griffith (1982). • Let X have an exponential distribution with mean 1, and let ~p(-) be a strictly increasing differentiable (except perhaps at ml and rn2) function on I0, ce). Further, if ~(.) is convex on [0, tal), linear on (ml, m2) and concave on (m2, ce) (where possibly ml -- m2), then ~p(X) has a BFR. Reliability and stochastic mechanisms.

• Series system (competing risk model). Suppose we have a series system of two independent components. It is well known that the failure rate of the system is simply the sum of the two component failure rates. If one of them has an IFR distribution and the other has a D F R distribution, then the system lifetime may have a bathtub-shaped failure rate. Models obtained by Murthy et al. (1973), Canfield and Borgman (1975), Gaver and Acar (1979) are of this type. • Stochastic failure models. Consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a homogeneous Poisson process, under appropriate conditions on the probability of surviving a given number of shocks, Mitra and Basu (1996b) have shown that the lifetime of the device has a BFR. I D M R L classes.

Also known as upside-down bathtub mean residual life distributions. A bathtub shaped failure rate distribution may arise via an I D M R L model. • Stochastic differential equation models and population abundance distributions. • Mixture. gamma mixture, Weibull mixture (different location parameters), mixture of three distributions (Krohn, 1969). Sectional models.

• Piecewise linear in three regions (Shooman, 1969), Colvert and Boardman (1976), Jaisingh et al. (1987). For example,

{

sl - qit,

h(t) =

0 < t _< tl,

~2, g 2 -}-

tl < t < t2,

q2(t

--

t2),

t > t2 ,

Bathtub-shaped failure rate life distributions

83

subject to the conditions tl = (el - e 2 ) / t / l , el > e2 > 0, t/1 , g/2 > 0. Other sectional models that give rise to BFR distributions are Jiang and Murthy (1997b), Murthy and Jiang (1997). Polynomial o f finite order. • Jaisingh et al. (1987) and Shooman (1968) suggested a polynomial of finiteorder failure model: h(t) = ao + alt + • •. + ant n. As the constants ai, i = 0 , . . . , n may be positive or negative, bathtub shapes can be achieved. T T T transformation. • See Kunitz (1989) and Haupt and Schabe (1997) for details. Truncation o f D F R distribution. • Schabe (1994) has constructed bathtub shaped failure rate distributions from decreasing failure rate distributions by truncations.

6. Tests of exponentiality versus bathtub distribution 6.1. Test based on total time on test ( T T T ) transform

Bergman (1979) suggested a test based on the total time on test (TTTT) transform for testing exponentiality against bathtub shaped distributions. Aarset (1985) derived the exact distribution of this test under the null hypothesis of exponentiality. Aarset (1987) also derived another test that is based on T T T plot which is equivalent to the well-known Cramer-von-Mises test statistic. A Monte Carlo power comparison of the two tests are performed by Kunitz 1989). See also Xie (1988). More recently, see Haupt and Schabe (1997). Formally, the T T T transform is defined as: HF1(t) =

B

F-l(t)

F(u)du, and

(6.1)

d0

Scaled T T T transform: OF(t) = H ; 1(t)/HF 1(1) .

(6.2)

Now d 1 1 ~ttH~ (t) -- h(F_l(t) )

Since F is increasing in t, it follows that F -1 (t) is also increasing. We note that as 1 OF(t) = h(F I(t))HFI(1) '

it is obvious that OF is concave for F being IFR and convex for F being DFR. For a bathtub shape distribution, we expect the T T T transform curve to have an s-shape, that is we anticipate the T T T plot to lie below the 45°-line in its leftmost part and above the line in its rightmost part (Aarset, 1987; Kunitz, 1989). The curves in Figure 3 summarize the situations.

84


Fig. 3. Plot of ~ße-against time t.

As pointed out by Haupt and Schabe (1997), such an s-shaped curve need not cross the diagonal line. An example can be given by the following function: for 0 0

has bathtub shape. However, the inclusion is possible when an additional condition is imposed upon the BFR class. THEOREM 3 (Gupta and Akam, 1995b). I f F c {BFRs} and that h(0) > 1/#, then F c {IDMRLs}. That is, {BFRs} - {h(0) < 1/#} C {IDMRLs}. (Note: {h(0) < 1//~} = {all the failure rate functionsthat have h(0) < 1//@)

9.2. Bathtub-shaped failure rate and decreasing percentile residual life function The "c~-percentile residual life function" (c~-percentile RLF) was first defined by Haines and Singpurwalla (1974). Joe and Proschan (1984) show that this function may be expressed as q~,~(t) = F 1(1 - (1 - ~)ff(t)) .

(9.3)

90


A distribution is DFRL-~, if and only if for some c~, 0 < c~ < 1,q~,F(t) decreases in t. Launer (1993) has shown that a distribution with a bathtub-shaped failure rate, h(t), is DPRL-c~ for all ~0 < Œ< 1 for some e0 > 0, provided there exists a to with h(to) >_ h(O). 9.3. Relationships among N W B U E ,

B F R and I D M R L

classes

DEFINITION 7 (Mitra and Basu, 1994). A life distribution F having finite mean # is said to be 'new worse then better than used in expectation' (NWBUE) if there exists a point to such that _> # #(t)],_t0 .

(9 ,4)

THEOREM 4 (Mitra and Basu, 1994). Let F be a continuous and strictly increasing life distribution. I f F is B F R with mean #, then it is N W B U E . Notationally, {BFR} C {NWBUE}. The converse of the above theorem is false as can be seen from the following example: B(t) =

{

e -t, e - l ~ ta,

((ù - g + 1)),

t/8exp

0 _ < t < 1, 1 Me 1) Y, then >d(p) Y. any of the statements in Theorem 4.1 holds true and X >M0,-2) Y, then >M(p-1) Yany of the statements in Theorem 4.1 holds true and X >Me 2) Y, then >te) Y'

A strong bearing of second-order stochastic dominance on harmonic new better (worse) than used ( H N B U E / H N W U E ) class may be observed. In fact a life

114

A. Chatterjeeand S. P. Mukherjee

distribution with finite mean is H N B U E / H N W U E provided it is dominated by (dominates) an exponential density having the same mean in second order stochastic (d(2)) sense. An extension of this observation to the d(p) case may be of interest, for which the following definition is proposed. DEFINITION 4.11. A life distribution for the random variable X with reliability function R(x,X) is said to be generalized HNBUE (HNWUE) ofpth-order provided the harmonic mean of the MRL function over (0,x) of R(p)(x,X) is dominated by (dominates) the corresponding arithmetic mean of R(p)(x,X), for every hOrt-negative integer p. After certain routine calculations an alternative definition may be proposed as follows. DEFINITION 4.11 (a). R is generalized H N W U E of pth-order provided, for all positive integer p and non-negative x

R(P+I)(x,X) >_#p(X)exp(-pX#(p_l)(X)/#p(X))/p! (The dual version is obtained by reversing the inequality). It is obvious that putting p = 1 the usual H N B U E / H N W U E class may be generated. The strong bearing among generalized H N B U E / H N W U E class and d(p) dominance may be stated in form of the following theorem. THEOREM 4.3. R is generalized HNBUE (HNWUE) of pth-order provided it is dominated by (dominates) an exponential density having the same mean in (p + 1)th-order stochastic sense. PROOF. The proof follows by considering the expression for R(p+I)(x,X) when X has exponential density having the same mean as R and applying Definition 4.1 to generate the condition given in Definition 4.11 (a). 5. Closure of generalized order relations

Preservation of various dominance relations under several reliability operations has been examined by Mukherjee and Chatterjee (1991a, b), Boland et al. (1994), Shaked and Shantikumar (1994) among others. As mentioned in Stoyan (1983) the following reliability operations are of wide interest in the context of dominance relations or aging properties: (i) (ii) (iii) (iv) (v) (vi) (vii)

Convolution. Mixture distributions. Formation of coherent system. Poisson shock model. Laplace transformation. Weak convergence. Monotonic transformation.

Equilibriumdistribution- its role in reliabilitytheory

115

In what follows, few such reliability operations will be considered to establish the closure properties of the generalized dominance relations. The rest will be taken up in later communications.

5.1. Preservation under mixture of distributions The preservation of likelihood ratio (LR), failure rate (FR) and stochastic (d) dominances under mixtures has been considered by Mukherjee and Chatterjee (1991a). Later an elegant proof with respect to FR - case has been provided by Boland et al. (1994). A detailed discussion in this regard may be available in Shaked and Shantikumar (1994). In continuation Nanda et al. (1996a) considered the closure of the generalized dominance relations under mixture distributions. The basic problem along with the results will be discussed now. Let the non-negative continuous random variables Xo and Yo respectively have reliability functions R(x,Xo) and R(x, Yo) where 0 is guided by a random mechanism having distribution function G(O). Let the compound or mixture random variables which are again non-negative and continuous based on Xo and Yo be denoted as X and Y having reliability functions R(x,X) and R(x, Y), where

R(x,X) = I R(x'X°)dG(O)' R(x, Y) = / R(x, Yo)dG(O) Consider the reliability functions of the pth-order equilibrium distribution corresponding to Xo and Yo as Rco)(x,Xo), and R(p)(X, Yo), respectively. The question that will be addressed may be stated in terms of the following theorem. THEOREM 5.1. (i) IfXol >p-FR YO2,for all 01 and 02 in the support of 0; then under the above set up X >p-FR Y. (il) IfXol >p-sT Yo2,for all 01 and 02 in the support of 0; then under the above set up X >p-ST Y. (iii) IfX01 >p-CXX02, for all 01 and 02 in the support of 0; then under the above set up X >p-CX Y. (iv) IfX01 >p-CV Yo2,for all 01 and 02 in the support of 0; then under the above set up X >p-CV Y. (Although Nanda et al. (op. cit.) considered two different 01 and 02 having same probability model G(O), for all practical purposes they may be taken to be equal.) Before proving the theorem an important lemma has been proved by Nanda et al. (1996a) which is as follows. LEMMA 5.1. For non-negative integer p, (i) R(p)(x,X) = f(-oo,oo)tip-1 (O)R(p)(x,Xo)dG(O),

116 (ii)

A. ChatteJjeeandS. P. Mukherjee

7p(X) = f(-~,o~)tip-l(O)'/p(Xo)dG(O), where tip(O)= HP=oTj(Xo)/IIP_oTj(X) with by convention/3_ 1(0) = 1.

PROOF. The proof follows by induction on p, starting with p = 1 and invoking Fubini's theorem for changing the order of integration. Proof of the theorem.

(i)

~01>p-FR1102 e=~ for

(d/dx)(R(p)(x,Xol)/R(p)(x, Y02)) ~" 0 [R(p)(x,Xol)R(p U(x, YO2))]/~/(p_I(YO2) all 01 and 02,

>- [Æ~-ll(x,Xol)R~l(x, Y02))]/~p ~(x0~) .

FIP=oTj(Yo)/IIP_oTj(y)

Define tip(0) : with by convention ti*l (0) = 1. Multiplication of both sides of the inequality by tip-1 (01)tip_l (02) and then application of expectation operator chronologically with respect to 01 and 02 yields

[/_oo,~)[tip-l(O)R(p-1)(x,Yo)/Tp I(Yo)]dG(O)]

~I/~ ~,~/~; l(0/~~~(~,~0/d~C0/J_>0 Observe that the left side of the above inequality is the numerator of the derivative of

[a~(-~,~)tiP-l(O)R(p)(X'X°)dG(O)]/[/( oc,oo)fl;-'(Õ)R(p)(X'Y°)dG(Õ)1 = R~I (x,X)/R~~(x, Y) ,

(by Lemma 5.1) with respect to x indicating that it is non-decreasing in x, which proves that X >p-FR Y.

Equilibrium distribution - its role in reliability theory

(ii)

Xol >p-ST Yo2

~

for all 0 and 02,

117

R(p)(x,Xol)/R(p)(x,Iio2)>_1

~:} ])p_l(Y02).f[x,e~)R(pl)(t,Xol)dt1 ~ 'p l(XO1)

f[x,~)R(Pl)(t'Y°2)dtI "

Multiplication of both sides of the inequality by ~p_2(01) • ~p_2(02) and then application of expectation operator chronologically with respect to 01 and 02 yields

>-I.f_~,~)flp2(O)Tp-l(X°)dG(O)l/[f(_~,o~)~*p-2(O)Tp-~(Y°)dG(O)1 " Interchanging the order of integration both in numerator and denominator of the left side of the inequality and invoking Lemma 5.1

ùf[x,oo)R(p1)(t,X)dt/.~~,oo)R(p_l)(t,Y)dt >_Ts_l(X)/Ts_l(Y), assuringX >p_sTY . For CX and CV orderings the basic idea of the proofs and the mode of derivation are almost similar and hence are omitted. For details reference may be made to Nanda et al. (op. cit.).

5.2. Characterization through Laplace transform The characterization of the usual dominance relations (particularly LR, FR and d) under Laplace transformation has been extensively studied by Kebir (1994), Shaked and Shantikumar (1994). Vinogradov (1973) has first studied characterization of the aging properties (more specifically IFR class) through Laplace Transformation. Later Block and Savits (1980) extended the theory to DMRL, IFRA, NBU, and NBUE classes. An obvious generalization of these results to characterize the p-FR and p-ST orderings as well as p-IFR, p-IFRA and p-NBU classes has been attempted and solved by Nanda (1995). A few definitions of discrete generalized orderings and generalized aging properties have been introduced in Fagiuoli and Pellerey (1993). Ler X and Y be two discrete non-negative random variables having probability mass functions {Pk} and {q»}, respectively, for non-negative integer k. Define for positive integer p, the discrete version of the equilibrium distributions as:

118

A. Chatterjee and S. P. Mukherjee

S(p)(k,X) =

S(p 1)(i,X)

S(p_l)(i»X )

with S(0)(k, X) =

Pk,

i=k+ I

S(p)(k, Y) =

S(p_0 (i , Y) i~k+ 1

/#_o

Se_l)(/, Y)

with S(0)(k, Y) = qk •

DEFINITION 5.1. (i) For two non-negative discrete random variables X and F, X is said to have pth-order failure rate dominance over Y (written X >p-VR I1) provided S(p)(k,X)/S(p)(k, Y) is non-decreasing in k. (ii) For two non-negative discrete random variables X and F, X is said to have stochastic dominance based on equilibrium distribution of order p over F (written X >p-ST Y) provided S(p)(k,X) > (S(p)(k, Y). Let for a non-negative random variable X with reliability function R(x,X), define the Laplace transform of X as ~(2,x)

=-

c¢ù(2,X) =

B0 ,~) exp(-Lv)

dR(x,X), 2 > 0 and denote

(-l)"(d'/(d2)")[(1 - +(2,X))/2]/n!, with

C~o(2,X) = (1 - 0(2,X))/2 and ~ù(2,x) = 2%

,(2,x)

.

Similar definition for another non-negative random variable Y having reliability function R(x, Y) is assumed to hold and is denoted by c¢,(4, Y), with the respective boundary condition and recurrence relation. Write for positive integer n, nonnegative x and any Bord measurable function defined on [0, oc), Fx (n, x) = 2 exp ( - ) « ) (2x)" -I / (n - 1)!

F;.g(n) = B ~) F2(n,x)g(x)dx with F~g(O) = 1

I

Kebir (1994) and Shaked and Shantikumar (1994) considered the following results proofs of which will be omitted as they are simple. LEMMA 5.2 (Kebir, 1994).

c¢ù(2,X) = [ F;~(n,x)R(x,X) JE0ùoc)

dx, for positive integer n

m

LEMMA 5.3 (Shaked and Shantikumar, 1994). ~, (2,X) and e, (2, Y) are proper discrete reliability functions .


119

Let N~x and N[ be two discrete random variables having respective reliability functions ~n(2,X) and ~~(2, Y). Invoking Lemma 5.3 it can be shown that:

F~R(p)(n,X) = f

F~(n,x)R(p)(x,X)dx and

r~~R(p)(n,r) = f

r~(n,x)Re)(x, r)dx

J[0ùoc) J[o,oo)

are also proper discrete reliability functions. The following lemmas will be used in the sequel and are proved in Nanda (1995). LEMMA 5.4. For all non-negative integer p,

(i) ~ rjR(p)(k,X) = 2~p(X)r~R(p+l)(n,X), k=n+l

(ii) ~ F~R(p)(k,X) = 1 + 2~p(X). k=0

Taking n = 0 a n d p = 1 in (i), orte can have E(Nf) = )LE(X) and E(N~) = 2E(Y). LEMMA 5.5. For all non-negative integers n and p, (i) S(p)(n, Nf) = FxR(p)(n + 1,X) and (ii) S(p)(n,N[) = F;R(p)(n + 1, Y). Three more lemmas will be required for convenience in discussion, which are: LEMMA 5.6 (Block and Savits, 1980). Ler x > 0 be a continuity point of any distribution function F. Let 2 = 2 ( x , n ) satisfy limn~~(n/2)=x and limn-~oo(1/2) = 0. Write dGn(t)= {2n+lt"exp(-2t)}dt/n!. Then Gù converges weakly to G, where G is degenerate at x. LEMMA 5.7 (Block and Savits, 1980). Let x > 0 be a continuity point ofR(p)(x,X). Let ,l = ,~(x, n) satisfy limn-~oo(n/2) = x and limn_~o~(1/2) = 0. Then for any nonnegative integer p, limù~oo F ~R(p)(n, X) = R(p) (x, X). LEMMA 5.8 (Block and Savits, 1980).

(d/d)OF;~R(p) (n,X) = (n/2) [F,~R(p)(n,X) - FxR(p)(n + 1, X)] . Block and Savits (1980) argued that for each 2 > 0, c~n(2,X) specifies a discrete reliability function corresponding to a specific cumulative damage model where X is the random threshold of the system and n being the number of shocks absorbed. It is to be noted that in a Poisson shock model the system is subject to a sequence of shocks occurring randomly as events in a Poisson process with intensity 2; each shock causes a random amount of damage to the system and the damages accumulate successively.

A. ChatterjeeandS. P. Mukherjee

120

Consider two systems which are subjected to sequence of shocks and the nature of shock be as above. Suppose X and Y be two random thresholds of the systems, respectively. Then the following theorem indicates that the necessary and sufficient condition that the thresholds are ordered according to p-FR (p-ST) sense is that the discrete lifetimes of the systems experienced by a sequence of exponentially spanned shocks are also ordered in p-FR (p-ST) sense. THEOREM 5.2. (i) (il)

X >p-VR Y ¢# Nf >p-VRN;Y for X >p-SW Y ¢:>Nf >p-sT Nf for

all positive 2, all positive 2.

PROOF. (i) ( ~ ) X >p-VR Y ¢V R@ (x,X)/R(p) (x, Y) is non-decreasing in x > 0, which means for some c > 0, R(p)(x,X) - cRco)(x, y) has atmost one change of sign and if it does occur, it occurs from below. Also it is well known that Poisson mass function P(2x, n - 1) is TP2 in (n,x), for x > 0. Thus by variation diminishing property of TP2 function (Karlin, 1968) B0,oo)Iexp (-2x)()«)"-~/(n - 1)!] [R(p)(x,X) - cR(p)(x, Y)] dx has atmost one change of sign and if it does occur, it occurs from below. Hence, for all positive integer n,

ff[o,oo)[exp(-2x)(2x)"-' / (n -1)'] R(p)(x,X)dx/ ~o

[eXp(-2x)()~x)" l/(n-1)']R(p)(x,Y)]dx

~CX3) L

is non-decreasing in n. Thus F2R(p)(n,X)/F~R(p)(n,Y) is non-decreasing in n and hence by Lemma 5.5 S(p)(n,N[r)/S(p)(n, Nf) is non-decreasing in n yielding N x >p-eR Nr, for all positive 2. ( ~ ) Note that the numerator of (d/d2)IraR~,)(~,x)/r~R~)(n, by invoking Lemma 5.8 can be written as

~)],

which reduces to

+ 1, Y) - r;R(p)(n, Y)r~R(p)(n + 1,X)] . (5.1) By virtue of IF,R~)(n,X)/F;,R(p)(n, Y)I being non-decreasing in n; (5.1) beeomes non-positive. Thus considering (5.1) as a funetion of 2, F~R(p)(n,X)/F;tR(p)(n,Y) is non-increasing in 2. Hence F,/xR¢o)(n,X)/F,/xR¢~)(n, Y) is non-decreasing in x. (n/,Z) [r)Rc~ ) (n,X)r)R(p)(n

Equilibriumdistribution its role in reliabilitytheory

121

Let 0 < xl < xa be two continuity points of both R(p)(,X) and R(p)(*, Y). Then

r,/xaR(p) (n,X)/rn/xaR(p)(n, Y) >_F~/x1R(p)(n,X)/Fn/«lR(p)(n, Y), for positive integer n. Taking limits as n approaches infinity and using Lemma 4.7,

R(p)(x2,X)/R(p)(X2, Y) > R(p)(Xl,X)/R(p)(X~, Y), for 0 < xl < x» As the set of discontinuity points of a reliability function is at most countable which means the set of continuity points is dense, R(p)(x,X)/R(p)(x, Y) is nondecreasing in x yielding X >p-FR Y. (il) ( ~ ) X >p-ST Y ~:* R(p)(X,X) > R(p)(x, Y) for all x _> 0. Multiplication of both sides of the above expression by F;.(n,x) and integration over x ~ I0, oc) yields F;R(p)(n,X) >_F~R(p)(n,Y), which by Lemma 5.5 reduces to S(p)(n, N x) >_ S(p)(n, NX), for all non-negative integer n, Thus Nf >p-ST N;Y, for all positive 2. ( ~ ) N~ >p-ST Nr, for all positive 2 ~ F~R(p)(n,X) > F~R(p)(n, Y) for all positive integer n. Let x (> 0) be continuity point of both R(p)(.X) and R(p)(*, Y). Then for each positive integer n, F~/~R(p)(n,X) > F~/~R(p)(n, Y). Taking limit as n approaches infinity on both sides of the above expression and using Lemma 5.7, R(p) (x,X) > R(p)(x, Y), for all continuity points x. Thus by the similar argument as of p-FR case, X >p-ST Y follows. Notes: (i) The above characterization is a generalization of the characterization of usual FR - dominance under Laplace Transform as has been stated in Kebir (1994) and Shaked and Shantikumar (1994). However, for p = 0, 2 and 3 the results corresponding to LR, M R L and VRL - dominance easily follow. (ii) The above characterization is a generalization of the characterization of usual d - dominance under Laplace Transform as has been stated in Kebir (1994) and Shaked and Shantikumar (1994). However, for p = 2 the result corresponding to H A M R - dominance easily follows. The characterization of the generalized aging classes in terms of Laplace transform may be stated in terms of the following theorem. THEOREM 5.3. (i) A random variable X is p-IFR iff F;R(p)(n,X)/F;R(p 1)(n,X) is non-increasing in n i.e. - l o g [F;~R~,~(n,X)] is convex in n, for positive integer n. (ii) A random variable" X lS'Lp-IFRA~/lff"~F,tR(p)(n,X)] 1/n.ls non-increasing in n i.e. - l o g IC;oR(p)(n,X)l is star-shaped in n, for positive integer n. (iii) A random variable X is p-NBU iff F~R(p)(m,X)FxR(p)(n,X) >_F~R(p) (m+ n,X) i.e. - l o g [F~R(p)(n,X)l is super-additive in n, for positive integer n. PROOF.

(i) ( ~ ) Consider x (> 0) and x + y (> 0) be two continuity points of both and F~R(p_I)(n,X). Define 2 = n i x and k=[ny/x]. Then lim,,_+o~(n + k)/2 = x +y. By Lemma 5.6, one can show that Gn+k-1 converges

F~R(p)(n,X)


122

weakly to a degenerate distribution with point of degeneracy x + y. Thus by Lemma 5.7, limn-~o~FxR(p)(n + k,X) = R(p)(X + y,X). As F;~R(pI (n,X)/rx R(p_l)(n,X) is non-increasing in n for positive integer n, by Lemma 5.8, F~R(p) (n,X)F~R(p_I)(n + k,X) - FxR(p)(n + k,X)F~R(p_I)(n,X) »_ O. Taking limit as n approaches infinity R(p) (x,X)R(p_l)(x q- y , X ) - R(p)(x q- y , X ) R ( p _ l ) ( x , X ) > 0 .

As the continuity points of reliäbility function are dense, the result follows immediately by noting that the above condition is equivalent to R(p)(x,X)/R(p_i)(x,X) is non-increasing in x i.e. -logR(p)(x,X) is convex in x. Thus X is p-IFR. ( ~ ) X is p-IFR ~ For any given c (> 0), R(p)(x,X) - cR(p_l)(x,X) has at most one change of sign and if a change does occur it occurs from above. As Poisson mass function P(2x, n - 1) is TP2 in (n,x), for positive integer n and x > 0, by variation diminishing property of TP2 function (Karlin, 1968)

B

,~)[exp(-)zc)(2x)n-l /(n - 1)!][R(p)(x,X) - cR(p_l)(x,X)]dcr

has atmost one change of sign and if it does occur, it occurs from above. This means FxR(p)(n,X) - cF~R(p_l)(n,X) has at most one change of sign and if it does occur, it occurs from above. Thus F~R(p)(n,X)/FxR(p_I)(n,X) is non-increasing in positive integer n which means -log[F~R(p)(n,X)] is convex in n, for positive integer n. (ii) Define c~,(2,X(p)) = f[0,~)F,~(n,x)R(x,X(p))dx, for positive integer n. Then

c~H(2,X(p) ) = J[Of,o~)[un exp(-2u) /n!]R(p) (u,X) du = 2([o [uH/n!]R(P)(u, Y) du, ùoo)

where R~)(u, Y) = exp(-)~u)R(p)(u,X)

= E[Y~p~l]/(n + l) ! • ( 3 ) Given that for positive integers n and k for all positive 2,

Taking x (> 0) and x + y (> 0) as the continuity points of R(p)(n,X) and using the same type of argument as used in (i), as n approaches infinity; the above expression reduces to

R~)(x,X) > [Re)(z +y,X)] ~/(~+~) .


123

Since this is true for all continuity points x of R(p)(x,X) and the continuity points are dense, the above expression reducës to [R(p)(x,X)l l/x is non-increasing in x, which means - l o g R(p)(x,X) is star-shaped in x. Thus X is p-IFRA. ( ~ ) Observe that R(p)(, Y) is I F R A iff R(p)(,X) is I F R A and X is p-IFRA means X(p) is IFRA. Thus by using Corollary 6.5 of Barlow and Proschan (1975), X is p-IFRA implies: 1)!} l/(n+l)

is non-increasing in n, whichmeans

{Œù(2,X(p))} 1/("+1) is non-increasing in n, which further gives

[F;,R(p)(n,X)I 1/, is non-increasing in n, which is equivalent to - log[F2R(p)(n,X)l is star-shaped in n for positive integer n (iii) ( 3 ) Given that for positive integers n and k, F2R(p)(n,X)F;~R(p) (k,X) >_F~R(p)(n + k,X). Taking limit as n approaches infinity and by similar argument as in (i) or (ii), it follows that: [R(p)(x,X)l [R(p)(y,X)] >_ R(p)(x +y,X), for positive x and y, which means -log[R(p)(x,X)] is super-additive in x. Thus X is p-NBU. ( ~ ) As in (ii) Re)(,, Y) is NBU iff R(p)(, X) is NBU and X is p-NBU means X(p) is NBU. Thus X is p-NBU implies {E[X;~l]/(m@

l),}{E[X;~l]/(n-~ -

m+n+l]/( m -~-n Jc- 1)!} 1)'} > { E XI (p)

This yields -log[F;R(p)(n,X)l is super-additive n, for positive integer n. Notes:

(i) (A) For p = 1, the characterization of I F R class as stated in Vinogradov (1973) follows. The condition is - log F;R(n,X) is convex in n (positive), for all positive 2. (B) For p = 2, the characterization of D M R L class as stated in Block and Savits (1980) follows. The condition is ~~=n FxR(k,X)/FxR(n, X) is nonincreasing in n (positive), for all positive 2. (c) For p = 3, the characterization of DVRL class follows. The condition is ~k~=n F;R(2)(k,X)/F;~R(2)(n,X) is non-increasing in n (positive), for all positive 2. (ii) For p = 1, the characterization of I F R A class as stated in Block and Savits (1980) follows. The condition is -log[FxR(n,X)] is star-shaped in n (positive), for all positive 2. (iii) For p = 1, the characterization of NBU class as stated in Block and Savits (1980) follows. The condition is -log[FxR(n,X)] is super-additive in n (positive), for all positive 2.

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6. Equilibrinm distribution in higher dimension So far the discussion is concerned with a non-negative random variable X and its equilibrium distribution of higher orders. In this section the multivariate version of equilibrium distribution will be proposed with regard to the n-dimensional random vector X = ( X l , ~ , . . ,Xn) I defined over the non-negative orthant of ndimensional real space denoted as IP,+. The extension of the univariate concept to multivariate case was first introduced by Mukherjee and Chatterjee (1991b) while establishing certain closure properties of two multivariate aging classes proposed by Johnson and Kotz (1975), Marshall (1975), Zahedi (1985), and Arnold and Zahedi (1988). For the sake of completeness the definitions are cited below. DEFINITION 6.1. Let R(x) = P[X > x] be the reliability function of the n-dimensional random vector X defined over IR+. The extended real vector b = (bi, b 2 , . . , b,,)' is called the supremum of the support (sos) of R provided bj = inf{xc[0,cm)}{x : Rj(x) = 1},

j = l(1)n ,

where Rj(x) is the j t h univariate marginal of R. It is apparent that some or all of the components of b may be ec. The following two multivariate ageing classes are proposed for X, a random vector defined over IR,+, having reliability function R. Also let b denote the sos of R. DEFINITION 6.2. Define the Borel measurable function on IP,+ as A(x, X) = ()Vl (x, X), )v2(x, X ) , . . , ~.n(x, X)', where for all x c IR+ and x x / » ) / R & A x / j ) > x/») , where f j (xjIXü)> x0) ) is the conditional density of Xj given X 0 ) > x0) and

Rj(xjIX(i ) > x0) ) = P[Xj > xjIX(i ) > x(j)] with x(j) = (Xl,X2,.. ,xj_~,xj+l,... ,x,,)'. Johnson and Kotz (1975) as well as Marshall (1975) called A(x,X) as the hazard gradient vector. It is to be noted that on the set {x: P[X > x] = 0} which is equivalent to the set {x:R(x, X) = 0}, the function may be defined in an arbitrary fashion. DEFINITION 6.3. Define the Borel measurable function on IR+ as r(x, X) = (rl (x, X), r2(x, X ) , . . ,

r,(x, X))',

where for all x E IR+ and x x)

=[

«[,j ùcm)

R(tj, x(i), X)dtj/R(x, X),

where (tj, x0) ) means (Xl,X2,.. ,xj 1, tj,xj+l,...

,xn)' •

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125

Zahedi (1985) and Arnold and Zahedi (1988) called r(x, X) as the multivariate mean remaining life function. Here also it is to be noted that on the set {x:P[X > x] = 0} which is equivalent to the set {x:R(x,X) = 0}, the function may be defined in an arbitrary fashion. With the above definition of A(x, X) and r(x, X) the following two multivariate aging classes may be proposed. DEFINITION 6.4 (Johnson and Kotz, 1975; Marshall, 1975). A reliability function R (x, X) or the corresponding random vector X is said to be vector multivariate increasing (decreasing) hazard rate (VMIHR (VMDHR)) distribution provided for all x E IP,+ and x < b (lexicographically) ~q(x) is non-decreasing (non-increasing) in xj. DEFINITION 6.5 (Zahedi, 1985; Arnold and Zahedi, 1988). A reliability function R (x, X) or the corresponding random vector X is said to be decreasing (increasing) multivariate mean remaining life of type 2 (DMMRL (IMMRL)-2) distribution provided for all x ~ IR+ and x < b (lexicographically) rj (x) is nondecreasing (non-increasing) in xj. Mukherjee and Chatterjee (1991b) have considered the multivariate reliability function according to the equilibrium distribution of R(x, X) as

R(2)(x, X) = f~;,~) R( tj, x(i), X)dtj / fo,~) R( tj, x(i), X)dtj ,

(6.1)

An interpretation of R(2)(x,X ) may be available in Gupta and Sankaran (1998). However in line of Section 5 the ¢oncept of higher order equilibrium distribution in the multivariate set up may be introduced by generating a sequence of equilibrium distributions corresponding to R(x, X) for all positive integer p as follows:

R(p+~)(x,X)= /xj,~)R(p)(tj, x(j),X)dtj/ f[o,~)R(p)(tj, x(j),X)dtj .

(6.2)

It is interesting to note that as in the univariate case here also R(p+l)(x, X) represents a proper multivariate reliability function in the usual sense. With the above definition of R(2)(x, X) Mukherjee and Chatterjee (op. cit.) proved a theorem establishing a relationship between V M I H R (VMDHR)-ness of R(2) and D M M R L (IMMRL)-2-ness of R. The same result with few extensions, only in bivariate set up has been cited in Gupta and Sankaran (1998). However, a general version of those results involving R(p+I)(x, X) may be cited in the following theorem. THEOREM 6.1. R(p+l)(x, X) is V M I H R (VMDHR) iff R(p)(x,X) is D M M R L (IMMRL)-2.

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126

PROOF. Observe that for j = l(1)n, 2jR~+~)(x, X) = jth component of the hazard gradient of R(p+l) (x, X).

= -((~/~xj)R~+~l (,,, x))/R¢o+l/(x, x)

=-{-IR(p)(Xj,X(j),X)/f[o,~)R(p)(tj,x(j),X)dtjl}/ Ifxj,oo)R(p)(tj,x(j),X)dtj/~,~)R(p)(tj,x(j)'X)dtj1 = R(p)(x, X)/flxj,oo)R(p)(tj, x(j),X)dtj = 1/rjR(p)(x,X), rjR

where• (x, X) is the jth component of the multivariate mean remaining life (p functlon o~ R(p)(x, X). Hence, for j = l(1)n, on IP,+, 2jRe+~/(x, X) non-increasing (non-decreasing) in (x, X) non-decreasing (non-increasing) in The concept of hazard gradient dominance (a generalization of failure rate ordering as stated in Ross, 1983) and multivariate mean remaining life dominance (a generalization of mean remaining life ordering as stated in Gupta and Kirmani, 1987) between two non-negative random vectors X and Y has been introduced by Mukherjee and Chatterjee (1993). Moreover, the concept of multivariate stochastic dominance based on reliability function was discussed in Marshall and Olkin (1979). The definitions involving two non-negative random vectors X and Y are restated for the sake of continuity in discussion.

xj e=~rjR~,I

xj.

DEFINITION 6.6 (Mukherjee and Chatterjee, 1993). X is said to have hazard gradient dominance over Y (written as X >HG Y) provided for all x ~ IR+ and x A(x, X), where the inequality is valid lexicographically . DEFINITION 6.7 (Mukherjee and Chatterjee, 1993). X is said to have multivariate mean remaining life dominance over Y (written as X >MMRL Y) provided for all x E IR+ and x r(x, Y), where the inequality is valid lexicographically . DEFINITION 6.8 (Marshall and Olkin, 1979). X is said to have multivariate stochastic dominance over Y (written as X >x Y) provided for all x c IR+ and x R(,,, Y)

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127

With the introduction of higher-order equilibrium distribution in multivariate set up as stated in 6.2 the notions of hazard gradient vector and multivariare mean remaining life vector may be extended as follows, while X and b are as before. DEFINITION 6.9. Define the Borel measurable function on IR+ as Aco+l)(x , X) = (2(p+X)l(x,X), 2(p+l)2(x,X),.. ,2(p+un(x,X))', where for all x c IR+ and x x] = O} which is equivalent to the set {x : R(p) (x, X) = 0}, the function may be defined in an arbitrary fashion, DEFINITION 6.10. Define the Borel measurable function on IR+ as

r~+~)(, x) = (~~+1)~(", x), ~~+~)~(x, x ) , . . , r~+~)~(x, X))' for all x q IP.+ and x < b (lexicographically) , r(p+l)j(x, X) = ff[i.j,oc)R(p)(tj, x(j), X)dtj/R(p)(x, X) r(p+l) (x, X) may be called as the multivariate mean remaining life vector ofpthorder. Here also it is to be noted that on the set {x : P[X(p) > x] = 0} which is equivalent to the set {x:R(p)(X, X) = 0}, the function may be defined in an arbitrary fashion. With the above definition of R(p) (x, X), A(p)(x, X) and r(p) (x, X) in line of definitions of Section 4 the following multivariate dominance relations of pth-order may be proposed between two random vectors X and Y defined over IR+. Let b denote the common sos of the reliability functions corresponding to X and Y. DEFINITION 6.1 1. X is said to have pth-order hazard gradient dominance over Y (written as X >p-HG Y) provided for all x E IR+ and x A(p)(x, X), where the inequality is valid lexicographically .

DEFINITION 6.12. X is said to have pth-order multivariate mean remaining life dominance over Y (written as X >p-MMRL Y) provided for all x C IR+ and x _ r(p)(x, Y), where the inequality is valid lexicographically .

DEFINITION 6.13. X is said to have pth-order multivariate stochastic dominance over Y (written as X >p-x Y) provided for all x E IR+ and x (R(~)(x,Y)

128

A. Chatterjeeand S. P. Mukherjee

Before elaborating certain issues concerning the higher order multivariate equilibrium distribution, consider the random vector of the residual life beyond t as Xt(p) corresponding to X(p) the random vector having reliability function R(p) (x, X). Ler its reliability function be denoted by Rt(p)(x, X). Then

Rt(p>(x, x) -- P[x,(p) > x] = P[X(p) > x + tIX(p) > t] = R(p)(x + t, X)/R(p)(t, X) . Moreover higher-order representations of various aging properties as introduced by Buchanan and Singpurwalla (1977) may be defined as follows. DEFINITION 6.14. X is said to have pth-order multivariate increasing (decreasing) failure rate distribution - very strong (written X is p-MIFR (MDFR)-VS) provided for all x, t E IP,+ and x, t < b (lexicographically) R(p) (x + t, X)/R(P) (x, X) is non-increasing (non-decreasing) in x . DEFINITION 6.15. X is said to have pth-order multivariate new better (worse) than used distribution - very strong (written X is p-MNBU (MNWU)-VS) provided for all x, t E IP,+ and x, t R(p/(x + t, x) . The following theorems connect p-MIFR-VS and p-MNBU-VS with p-K in one hand and establish interrelations between p-HG, p - M M R L and p-K on the other. All the results as stated are generalizations of the results obtained in Mukherjee and Chatterjee (1993). THEOREM 6.2. (i) X is p-MIFR-VS provided for all x, y E IR+ and x < y ; « Xy(p) and vice versa. (ii) X is p-MNBU-VS provided for all y E IP,+ and y p-x Xy(p) and vice versa.

Proof. (i) Observe that for all x, y ~ IR,+ and x < y p-K Xy(p) ~ R(p)(x -~- t, X)/R(p)(x, X) > R(p)(y + t, X)/R(p)(y, X) ¢=~R(p)(x + t,X)/R(p)(x, X) is non-increasing in x ¢=~X is p-MIFR-VS (il) Observe that for all x, y E IR+ and x < y p-K Xy(p) ~ R(p)(x,X) ~_ R(p)(y ~- x, X)/R(p) (y, X)

R(p) (x, X)R(p)(y, X) > R(p)(y + x, X) ~=~X is p-MNBU-VS . Consider the residual life of X(p) beyond a random cut-off point Y having distribution function G(y, Y) and denoting it by Xv(p) with the corresponding reliability function Ry(p)(x,X) = P[X(p) > x + YIX(p) > Y]

= f(y

R@(x + y, X)dG(y, Y ) / E N+ and y x _u y + t]X(p) > Y + t]

= P[X(P) > x + Y + t]/P[X(p) > Y + t] .

(6.3)

It will be shown that if X is p-MIFR VS then Xt(p) >p-X Xy+t(p)Corlsider Rt(p)(x, X) - Ry+t(p)(x, X) = R(p)(x -~- t, X)/R(p)(t, X) - R(p)(Y -- x -It, X)/R(P)(Y + t, X) which has the same sign as

f,

[R(p)(x + t, X)R(P)(y + t, X) ' E N + and y Ry+t(p) (x, X) ensuring (6.3). Putting t = 0 in (6.3), it is quite natural that X(p) >p-K Xy(p). Notes: (i) The converse of the Theorem 6.3 is not true, although that of Theorem 6.2 is evidently valid.

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A. Chatteljee and S. P. Mukherjee

(ii) By reversing the dominance relations and inequalities and by replacing nonincreasing by non-decreasing the results corresponding to the dual versions of Theorems 6.2 and 6.3 may be derived. (iii) By substituting x* = x.1, y* = y.1 and t* = t.1 where 1 is an n-dimensional unit vector in place of x, y and t in the respective cases, the extended results concerning 'S', 'W' and 'VW' versions as referred in Buchanan and Singpurwalla (1977) can be easily obtained. THEOREM 6.4. For two random vectors X, Y the following statements are equivalent for x, t c IP-+ and x, t e-rm Y. (il) R(p)(x, X)/R(p)(x, Y) is non-decreasing in every component of x. (iii) Xt(p) >p-X Yt(p). PROOF. Observe that for x, t E IP,+ and x, t < b (lexicographically) and for all integer j = 1(1)n,

(6~fixe) ln(R(p)(x, X)/R(P)(x, Y)) = (R(p)(x, Y)/R(p)(x, X)R~p)(x, Y)) [R(p)(x, Y)(6/&•)R(p)(x, X)

- e(p / (x, x)(a/ax:)e(p I (,,, Y)]

= -~~(p/+(x, x) - ,~(p>+(x,Y) Thus (ii) ~

(3/&/) ln(R(p)(x, X)/R(P)(x, Y)) > O,

for all integer j = l(1)n ¢~ 2(p)/(x, Y) _> 2(p)/(x, X), for x e IP-+ and x R(p)(x + t, X)/R(p)(x + t, Y) _> R(p)(t, X)/R(p)(t, Y) (ii). THEOREM 6.5. (i) X >p-HG Y ~ X >v-K Y and the converse is not generally true, (ii) X >p-H~ Y ~ X >p-MMRL Y and the converse is not generally true. PROOF.

O) X >p-HG Y A+)(~, X), where the inequality is valid lexicographically.

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its role in reliability theory

131

Integration of both sides over the piecewise smooth path in [0, t] (cfs. Block, 1977; Galambos and Kotz, 1978; Mukherjee and Chatterjee, 1988) yields - lnR(p)(t, Y) _> - lnR(p)(t, X), assuring X >p-K Y •

(ii) By virtue of Theorem 6.4, for all x, t E IR+ and x, t p-HG Y ¢:> R(p)(X + t,X)/R(p)(x,X) > R(p)(X + t,Y)/R(p)(x,Y) .

(6.») Taking t = (tj, 0.t(j)) in particular, 6.5 reduces to

,e~/(xj + t» x~» x)/R~/(x, x) >_R~/% + t» x~»,v)/R~/(x, Y) Integrating both sides with respect to

/xj

tj over the

range [0, oc) yields

R(p)(yj'x(j)'X)dyj/R(p)(x'X) >- ~xj R(p)(yj, x(j),Y)dyj/R(p)(x,Y) ** r(p)j(x,X) >_ r(p)j(x,Y)

for all integer j = l(1)n .

The reverse implications are not valid since in even for n = 1 and p = 1, the corresponding implications are not true. For counter examples Gupta and Kirmani (1987) may be referred. It is to be observed that for n = 1, that is the univariate set up, p - H G e=~p-FR; p-K ~=~ p-ST and p - M M R L

¢:~ p - M R L .

7. Equilibrium distribution in the context of repairable system

For a repairable system, let {S(n) : n = 1 , 2 , . . } be the time of the nth failure of the system such that 0 = S(0) < S(1) < ... < S(n). DefineX(n) = S(n)-S(n - 1), n = 1 , 2 , . . , as the time between ( n - 1)th and nth failure. Thus S(n) will be a renewal process, corresponding to which let N(t) being a counting renewal process.

Define Xs(n_l),..,s(1)(n ) as the r.v. X(n) given S(n-1)=s(n-1),.., S(1) = s(1), where s ( 1 ) , .. ,s(n - 1) are real numbers, n = 2, 3 , . . , and for n = 1 this is trivially taken as X(1), indicating lifetime of the new system. Ebrahimi (1989) defined the counting process N(t) consisting of inter arrival times X(1), X ( 2 ) , . . . , is improving provided Xs(j-l),..,s(1) (J') >ST Xs(i-l),..,s(1)

(i), for

j>i>

every positive integer

1; 0 < s ( 1 ) < s ( 2 ) < . . . < s ( j - 1 )

.

Replacing ST by LR, FR, M R L and E, Deshpande and Singh (1995) and, by VRL and CX, Bagai and Jain (1994) extended the concepts of system improve-

132


ment (deterioration). As p-FR and p-ST are more generalized version of usual order relations, replacing ST by p-FR and p-ST the definitions and results of Ebrahimi (1989), Deshpande and Singh (1995) and Bagai and Jain (1994) may be generalized and that has been attempted by Nanda (1997). It is obvious that putting p = 0, 1,2 and 3 the results corresponding to LR, FR, MRL and VRL follow. The concept and notion of minimal repair is of wide interest among reliability practitioners. For a repairable system, by minimal repair it is meant that the failed system is restored to a state of functioning which is as good as the condition just prior to failure. Thus, by definition, a minimal repair enables the system to work keeping the failure rate unchanged. Thus if the system fails at time 't' and undergoes minimal repair, the reliability function Rt(x) of the minimally repaired system is given by

Rt(x,X) = R(x + t,X(1))/R(t,X(1)) ,

(7.1)

where R ( , X ( 1 ) ) is the reliability function of X(1). Denote by N(t), the counting process giving the number of minimal repairs upto 't'. Let, for notational convenience, X(1) be Y~ and X~(,)......(1)(n + 1) be 11ù+1,for n = 1,2, 3 , . . . Yj's are assumed to be absolutely continuous non-negative random variables having reliability fnnction R(x, Yj). Considering the higher-order equilibrium distributions corresponding to R(x, Yj) as RCo) (x, Yj), for non-negative integers p, Nanda (1997) established the following relationship connecting R(p)(x,Y~+l) and

Rc~)(x, r~). RESULT 7.1.

R(p)(x, Y,+I) = [70(Y1)?l(Y1)...~p-l(Y1))/ Y0(Yn+I)YI (Yn+l).-- 7p-1 (Yn+l)l × [Rc~) (x + s(n), r,)/Rco ~(s(n), ~r,)] with 7-1(,) = 1. PRoov. The proof is simple and follows by induction on p. The following theorem gives the effect of aging of X(1) on the improvement (deterioration) of a system undergoing minimal repairs at failures and is proved in Nanda (1997). THEOREM 7.1. For any non-negative integerp, the stochastic process {N(t):t > 0} generated by a minimal repair policy is improving (deteriorating) in: (i) p-FR sense iff R(x,X(1)) is p-DFR (IFR), (ii) p-ST sense iff R(x,X(1)) is p-DFR (IFR). PROOF. Consider the proof only in the case of 'improving'. For °deterioration' the proof holds with obvious modifications.

Equilibrium distribution its role in reliability theory

133

(i) The system is improving in p-FR sense means for all n = 1 , 2 , . . , and nonnegative x,

R(p) (x, Yn+l)/R(p)(x, Yù) is non-decreasing in x,

Yn+l >p-FR Yn

R~,)(x + s(n), rl)/RC~~(x + ,(n -

1), Y1)

is non-decreasing in x by Result 7. 1.

RCo)(x + t, Y1)/R(p)(X, Y1) is non-decreasing in x, t > 0.

¢:> Y1 is p-DFR .

(ii) The system is improving in p-ST sense means for all n = 1 , 2 , . . , and nonnegative x, Y~+I >p-ST Y~ g:} R(p)(X, Yn+l)/R(p)(O, Yn+l) ~ (R(p)(X, Yn)/R(p)(O, Yn)

¢:> Rcõ)(x + s(n), Y1)/R(p)(X + s(n - 1), Y1) >_R(p)(S(n), Y1)/R(p)(S(n - 1)Yl), by Result 7.1. R(p)(X + t, Y1)/R(p)(X, Y1) is non-decreasing in x, t > 0. *:> 111 is p-DFR . Notes: (i) The point process {N(t) : t _> 0} generated by a minimal repair policy is improving (deteriorating) in: (a) (b) (c) (d)

LR sense iffR(x,X(1)) is D L R (ILR), FR sense iffR(x,X(1)) is D F R (IFR), M R L sense iffR(x,X(1)) is I M R L (DMRL), VRL sense iffR(x,X(1)) is IVRL (DVRL).

The proof follows from the above theorem part (i) by putting p = 0, 1,2 and 3, respectively. It is to be observed that (a) (c) have been proved in Deshpande and Singh (1995) while (d) has been proved by Bagai and Jain (1994). (ii) The point process {N(t) : t > 0} generated by a minimal repair policy is improving (deteriorating) in: (a) W L R sense iffR(x,X(1)) is D L R (ILR), (b) ST sense iffR(x,X(1)) is D F R (IFR), (c) H A M R sense iff R(x,X(1)) is I M R L (DMRL). The proof follows from the above theorem part (ii) by putting p = 0, 1 and 2, respectively. As before here also it may be noted that (b) has been proved in Deshpande and Singh (1995).

A. Ckatteljee and S. P. Mukherjee

134

8. Characterization of exponential distribution Characterization of exponential distribution is abundant in the literature. Mukherjee and Chatterjee (1990) proposed some characterization resnlts based on integral equations involving RC°)(x,X). In what follows two of those results are restated in terms equilibrium distributions of order p i.e. R(p)(x,X), associated •with alternative and more elegant proof. THEOREM 8.1.

rR~+~)(x,X) = rR(p)(x,X) ¢* X has exponential density . PROOF. Observing rR(p)(x,X)=I/2R(p+l)(x,X), the above equality yields rR(p+l) (x,X) 2R(p+l)(x,X) = 1. This is the characterizing equation of R(p+l)(x,X) = exp(-x/c0, c~> 0. In view of Result 4.1

B

,oc)

R(p)(t,X)dt = (~(p+l)(X)/(p-- 1)p(p)(X)) exp(-x/~) .

Differentiating both sides with respect to x, yields R(p) ( x , X ) : (#(p+l)(X)//°:(P -}- 1)p(p)(X)) exp(-x/c~) . As RCo)(x,X ) is a proper reliability function, R(p)(O,X) = 1 gives ~ = #(p+l)(X)/ (p + 1)p(p)(X). Thus R(p)(x,X) = exp(-x/(p(p+l)(X)/(p + 1)p(p)(X))). The exponentiality of X follows immediately. THEOREM 8.2.

rR(p+i) (x,X)R(p+I) (x,X) = fR(p) (X, X)R(p) (x,X) OezX has exponential density . PROOF. By the same observation as in Theorem 8.1, the above equality yields rR(p+l) (x,X),~R(p+l)(x,X) = R(p)(x,X)/R(p+l ) (x, X ) = (~~l(x)/(p),~

l/(x));~~~+~/(~,x)

Thus ~~e+~/(~,x) = 1/x~~+~/(~,x)

=

~(~l(x)/~)~~

yielding

R(p+2)(x,X) = exp(-x/(#Co)(X)/(p)#(p_ U(X))) . The exponentiality of X follows immediately.

~)(x) ,

.


135

Characterization of several non-exponential distribution e.g. Pareto, L o m a x and Finite Range has been attempted by several authors in univariate set up. In the bivariate set up similar characterizations have been attempted by G u p t a and Sankaran (1998). In fact they have attempted to extend the results to multivariate cases also. For details of the p r o o f and related articles reference may be made to G u p t a and Sankaran (1998).

9. Concluding remarks Although the present article tried to cover almost all important aspects of equilibrium distribution(s), it is in no way claimed to be exhaustive. There are several areas which could not be given due elaboration or attention so as to prevent the paper becoming unnecessarily lengthy. The following among others continue to remain open problems and deserve close attention in future. 1. To establish the closure or otherwise under other reliability operations (not considered earlier) of various dominance relations and aging classes based on equilibrium distributions of higher orders. 2. To establish various closure properties or otherwise under all reliability operations of dominance relations and aging classes based on equilibrium distributions in higher dimensions and of higher order. 3. To study whether FR, ST and M R L imply p - F R , p-ST and p - M R L (integer p > 1) respectively and if such relations hold, how to interpret them. 4. To solve appropriate inference problems mainly related to p - F R , pST and d(p) order relations on the one hand and p - I F R , p - I F R A and p - N B U on the other.

Aeknowledgement The authors are grateful to Professor A. P. Basu, Department of Statistics, University of Missouri - Columbia, USA and Professor J. V. Deshpande, Department of Statistics, University of Pune, India for m a n y helpful suggestions and comments towards preparation of this article.

Referenees Arnold, B. C. and H. Zahedi (1988). On multivariate mean remaining life functions. J. Multivariate Anal. 25, 1 9. Bagai, I. and K. Jain (1994). Improvement, deterioration and optimal replacement under age - replacement policy with minimal repair. IEEE Trans. Reliab. 43, 156 162. Block, H. W. (1977). Multivariate reliability classes. In Applications ofStatistics, pp. 79 88 (Ed. P. R. Krishnaiah). North-Holland, Amsterdam. Block, H. W. and T. H. Savits (1980). Laplace transforms for classes of life distributions. Ann. Prob. 8, 465-474. Boland, P. J., E. E1 Neweihi and F. Proschan (1994). Application of hazard rate orderings in reliability and order statistics. J. Appl. Prob. 31, 180-192.

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A. Chatterjee and S. P. Mukher/ee

Buchanan, H. B. and N. D. Singpurwalla (1977). Some stochastic characterization of multivariate survival. In Theory andApplication ofReliability, Vol. 1, pp. 32%348 (Eds. C. P. Tsokos and I. N. Shimi). Academic Press, New York. Chatterjee, A. (1993). Unpublished Ph. D. Thesis, University of Calcutta, Calcutta, India. Cox, D. R. (1962). Renewal Theory. Methuen, London. Deshpande, J. V., S. C. Kochar and H. Singh (1986). Aspects of positive aging. J. Appl. Prob. 23, 748758. Deshpande, J. V., H. Singh, I. Bagai and K. Jain (1990). Some partiai orders describing positive ageing. Commun. Stat. - Stoch. Models 6, 471-481. Deshpande, J. V. and H. Singh (1995). Optimal replacement of improving and deteriorating repairable systems. IEEE Trans. Reliab. 44, 500-504. Ebrahimi, N. (1989). How to define system improvement and deterioration for a repairable system. IEEE Trans. Reliab. 38, 214-217. Fagiuoli, E. and F. Pellery (1993). New partiaI orderings and applications. Naval Res. Logist. 40, 829842. Fagiuoli, E. and F. Pellery (1994). Preservation of certain classes of life distributions under Poisson shock model. J. Appl. Prob. 31, 458~465. Fishburn, P. C. (1980). Stochastic dominance and moments of distributions. Math. Oper. Res. 5, 94100. Fisher, R. A. (1934). The effect of methods of ascertainment upon the estimation of frequencies. Ann. Eugenics 6, 13-25. Galambos, J. and S. Kotz (1978). Characterizations of probability distributions. In Lecture notes in Mathematics, Vol. 675 (Eds. A. Doldand and B. Eckmans). Springer, Berlin. Gupta, R. C. and S. N. U. A. Kirmani (1987). On order relations between reliability measures. Commun. Stat.: Stoch. models 3, 149 156. Gupta, R. P. and P. G. Sankaran (1998). Bivariate equilibrium distribution and its applications to reliability. Commun. S t a t . - Theory Meth. 27, 385-394. Johnson, N. L. and S. Kotz (1975). A vector multivariate hazard rate. J. Multivariate Anal. 5, 53-66. Erratum 498. Karlin, S. (1968). Total Positivity, Vols. I and II. Stanford University Press, CA. Kebir, Y. (1994). Laplace transform characterization of probabilistic orderings. Prob. Eng. Inf. Sei. 8, 69-77. Marshall, A. W. (1975). Some comments on the hazard gradient. Stoch. Process Appl. 3, 293-300. MarshalI, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and Applications. Academic Press, New York. Mukherjee, S. P. and A. Chatterjee (1988). A new MIFRA class of life distributions. Calcutta Stat. Assoc. Bull. 37, 67 80. Mukherjee, S. P. and A. Chatterjee (1990). Some characterization of exponential law through integral equations. In Proceedings of the Second Biennial Conference of the A llahabad Mathematical Society, Symposium on Statistics, 1990, pp. 61 67. Mukherjee, S. P. and A. Chatterjee (1991 a). On some closure properties of dominance relations based on reliability measures. Calcutta Stat. Assoc. Bull. 40 (H. K. Nandi Memorial Volume), 325 342. Mukherjee, S. P. and A. Chatterjee (1991b). On some properties of multivariate aging classes. Prob. Eng. Inf Sci. 5, 523 534. Mukherjee, S. P. and A. Chatterjee (1992a). Stochastic dominance of higher orders and its applications. Commun. Stat.: Theory Meth. 21, 1977 1986. Mukherjee, S. P. and A. Chatterjee (1992b). Closure under convoiution of dominance relations. Calcutta Stat. Assoc. Bull. 42, 251-254. Mukherjee, S. P. and A. Chatterjee (1993). Multivariate stochastic dominance with some implications. In Advances in Reliability, pp. 281-290 (Ed. A. P. Basu). Elsevier, Amsterdam. Nanda, A. K. (1995). Stochastic order in terms of Laplace transform. Caleutta Stat. Assoc. Bull. 45, 195-201.

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its role in reliability theory

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Nanda, A. K. (1997). On improvement and deterioration of a repairable system. IAPQR Trans. 22, 107-113. Nanda, A. K., K. Jain and H. Singh (1996a). On closure of some partial orderings under mixtures. J. Appl. Prob. 33, 698 706. Nanda, A. K., K. Jain and H. Singh (1996b). Properties of moments for s-order equilibrium distributions. J. Appl. Prob. 33, 1108 1111. O'Brien, G. L. (1984). Stochastic dominance and moment inequalities. Math. Oper. Res. 9, 475 477. Rao, C. R. (1965). On discrete distributions arising out of methods of ascertainment. In Classical and Contagious Diserete Distributions, pp. 320-332 (Ed. G. P. Patil). Pergamon Press and Statistical Publishing Society, Calcutta. Ross, S. M. (1983). Stochastic Process. Wiley, New York. Shaked, M. and J. G. Shantikumar (1994). Stochastie Orders and Their Applieations. Academic Press, New York. Singh, H. (1989). On partial orderings of life distributions. Naval Res. Logist. 36, 103 110. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. (Ed. D. J. Daley). Wiley, New York. Van Zwet, W. R. (1964). Convex Transformations of Random variables. Mathematisch Centrum Amsterdam, Amsterdam. Vinogradov, O. P. (1973). The definition of distribution function with increasing hazard rate in terms of the Laplace transforms. Theory Prob. Appl. 18, 811-814. Whitt, W. (1985). The renewal process stationary excess operator. J. Appl. Prob. 22, 156-157. Zahedi, H. (1985). Some new classes of multivariate survival functions. J. Stat. Plann. Infer. 11, 171 188. Zygmund, A. (1959). Trignometric Series, Vol II. Cambridge University Press, New York.


Reliability and Hazard Based on Finite Mixture Models

Essam K. AL-Hussaini and Khalaf S. Sultan

1. Introduction

In recent years reliability has been formulated as the science of predicting, estimating or optimizing the probability of survival, the mean life, or more generally the life distribution of components or systems. Interest in reliability has been manifested by mathematicians, engineers, medical scientists, economists and those concerned with the environmental and life sciences. Such developments and interest have been the interplay between reliability and statistics. Thus, the theory of statistical inference plays an important role in life testing and reliability problems. This has led to many published papers and books, such as, Barlow and Proschan (1965, 1975, 1981), Nelson (1982) and Lawless (1982). Recently, the book edited by Balakrishnan (1995) surveyed different problems in reliability and life testing, for example: (i) Parametric models and inference, see Doganaksoy (1995), Nelson and Doganaksoy (1995), Balakrishnan and Chan (1995a, b), Smith (1995), Kulasekera and Nelson (1995), Johnson et al. (1995), Pefia (1995) and Bandyopadhyay and Basu (1995), (ii) Survival analysis and multivariate models, see Arnold (1995), Anderson (1995) and Ebrahimi (1995). Also, the book by Elsayed (1996) emphasized more recent developments in engineering reliability. Statistical models play an important role in the field of reliability, for example justification for using the exponential assumption in reliability applications can be found in the early works of Davis (1952), and Epstein and Sobel (1953). Further justification, in the form of theoretical arguments supporting the use of the exponential distribution as the failure law of complex equipments, can be found in books by Barlow and Proschan (1965, 1975, 1981). Recent studies in reliability estimation and applications based on exponential distribution have been given in Engelhardt (1995). Some properties of distributions with periodic failure rates are discussed by Prakasa Rao (1997). The class of Inverse Gaussian distributions is currently challenging and is being used quite commonly as a lifetime model in reliability studies. Because of this, several authors have paid special attention to inferential issues concerning these distributions. References may be found in books by Chhikara and Folks (1989) and Seshadri (1993, 1999). In this connec139

I40

E. K. AL-Hussaini and K. S. Sultan

tion, the book edited by Balakrishnan and Chen (1997) presents extensive discussions and tables for order statistics from these distributions. AL-Hussaini (1999a) obtained Bayesian prediction bounds for observables from a general class of life distributions which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), Pareto, beta, Gompertz and compound Gompertz distributions. He suggested a general class of priors and based his prediction on the one- and two-sample cases. Bayesian prediction bounds of the median of future observations from this general class of distributions have been investigated by AL-Hussaini (200la). The study of homogeneous populations with 'single component' distributions was the main concern of statisticians along history, although Newcomb (1886) and Pearson (1894) were two pioneers who approached heterogeneous populations with 'finite mixture distributions'. Newcomb (1886) applied the normal mixture as models for outliers, whereas Pearson (1894) used the method of moments to estimate the parameters of a finite mixture of two univariate normal components with unequal variances. With the advent of computing facilities, the study of heterogeneous populations, which is the case with many real world populations, attracted the interest of several researchers from about the middle of the 20th century. Titterington et al. (1985) classified the applications of finite mixture models to direct and indirect applications. Direct applications cover fisheries research (lengths of different types of fish, etc.), economics (wage and housing regressions, number of purchases, etc.), medicine (clinical test scores, age of onset of arthiritis, teeth pressures, cholesterol triglyceride data, chromosome association, plasma glucose level, blood pressure, clinical measurements, etc.), geology (palaentology: foraminifer diameters, Egyptian mandible sizes, sedimentology: reflectivity (caol types), limestone cross-bedding, sand grain sizes, metal concentrations, volcanic ash deposits, etc.), electrophorisis (absorption spectrum, protein concentrations, gas chromatograms, etc.), botany (pollent grains, flowering times, plant heights, etc.), agriculture (crop concentrations, barley yields, panicle lengths, etc.), zoology (bird speeds, egg counts, rat species skull sizes, dental distances, etc.), engineering (valve lifetimes, transmitter receiver lifetimes, bus failure times, telephone call lengths, laser lifetimes, reliability, etc.). Indirect applications cover the contaminated normal models, outliers and robustness, normal mixtures as checks on robustness (for example, see Andrews et al., 1972; David, 1981; Huber, 1981; Tiku et al., 1986; Arnold and Balakrishnan, 1989; Gastwirth and Cohen, 1970; Balakrishnan and Geyer, 1995; and Barnett and Lewis, 1993), Gaussian sums, cluster analysis (see, for example, McLachlan and Basford, 1988), modeling prior densities, the use in the empirical Bayes methods, kernel density estimation, approximations of mixture rnodels by non-mixture models, random variate generation, among others. Four books have been written on mixtures: Everitt and Hand (1981), Titterington et al. (1985), McLachlan and Basford (1988) and Lindsay (1995). The first book concentrated on the estimation of the parameters of finite mixtures of normal, exponential and other continuous components and finite mixtures of discrete components, using the graphical, moment and maximum likelihood

Reliability and hazard based on finite mixture models

141

methods. The second listed and discussed real world applications of finite mixtures, the methods of estimation of their parameters and sequential procedures. The third also discussed mixtures with normal components and estimation of parameters using the EM algorithm; see McLachlan and Krishnan (1997). Applications of mixture models to the two-way data sets were discussed, estimation of the mixing proportions and the assessment of mixture likelihood approach to clustering of the three-way data. The fourth book discussed, in addition to other topics, the non-parametric and semiparametric maximum likelihood estimation under mixture models. Estimation of the parameters based on finite mixture models were studied, mainly when the components belong to the same family. For example, the parameters of a mixture of normals were estimated by Pearson (1894), Day (1969) and Quandt and Ramsey (1978), among others. The parameters of a mixture of Inverse Gaussian components were estimated by Ahmad (1982) and Amoh (1983). Kao (1959) used the graphical method in estimating the parameters of a finite mixture of two Weibull components. AL-Hussaini and Ahmad (1984) obtained the information matrix for a mixture of two Inverse Gaussian distributions. ALHussaini and Fakhry (1995) characterized a finite mixture of a general class of k components that are used in life testing. AL-Hussaini and Osman (1997) obtained the median of a finite mixture of k components. AL-Hussaini et al. (1997) used parametric and non-parametric methods to estimate the reliability function P(Y < X) when each of X and Y is a finite mixture of lognormal components. AL-Hussaini et al. (1999) studied the finite mixture of two-component Gompertz lifetime model. AL-Hussaini (1999b) used the Bayesian method to predict observables under a mixture of two-exponential components model. Predictive densities of order statistics based on finite mixture models with general class components have been obtained by AL-Hussaini (2001b). Certain practical situations require finite mixtures with components that do not necessarily belong to the same family. For example, Davis (1952) investigated a normal-exponential mixed model, Mudhlokar et al. (1995) applied the exponential-Weibull model to the bus-motor-failure data presented in Davis (1952), Ashton (1971) applied the 'modified semi-Poisson' model to distributions for gaps in road traffic, Ashour (1985, 1987) studied the Weibull-exponential mixed model and ALHussaini and Abd-EL-Hakim (1989, 1990, 1992) the Inverse Gaussian Weibull model. Teicher (1960, 1961, 1963, 1967) introduced the concept of identifiability and developed a theory to identify mixtures. Other researchers who studied identifiability of mixtures, countable or finite mixtures are Medgyessy (1961), BarndorffNielson (1965), Patil and Bildikar (1966), Yokowitz and Spragins (1968), Tallis (1969), Rennie (1972), Chandra (1977), AL-Hussaini and Ahmad (1981), Ahmad and AL-Hussaini (1982), Tallis and Chesson (1982) and Ahmad (1988). Other references may be found in Titterington et al. (1985), McLachlan and Basford (1988) and Maritz and Lwin (1989).


142

2. Concepts, notation and definitions In this section, we introduce, some basic concepts, notation and definitions. These include reliability function, hazard rate function, mean residual lifetime and mixture of distributions. The instantaneous rate of failure or death at time t, given that an individual survives up till t is known in demography and actuarial science as the force of mortality. In the theory of reliability, it is called hazard rate, failure rate or agespecific failure rate. In extreme-value theory, it is known as the intensity rate and its reciprocal is termed Mills Ratio in economics, see, for example, Mann et al. (1974), Kalbfleisch and Prentice (1980), Elandt-Johnson and Johnson (1980), Nelson (1982) and Lawless (1982). We shall use the term hazarfl rate funetion (HRF), denoted by r(t) throughout this chapter. If T is a a non-negative random variable which has a (continuous) cumulative distribution function (cd0 F(t) and a corresponding probability density function (pdf) f(t), then the H R F is given, for t > 0, by

r(t) = lim ( P[t t] = 1 - F(t) ,

(2.2)

is known as the reliability (survivor) function (RF). Knowing any of the density, reliability or HRFs leads to the knowledge of the other two by using the relationships

R(t) = e x p [ - fotr(s)ds]

,

(2.3)

1 d

r(t) -

R(t) dt [R(t)] ,

f(t) = -R'(t)

.

(2.4) (2.5)

Examination of (2.3) indicates that r(t) is a non-negative function with fö r(s)ds < ec, for some u > 0 and f ö r(s)ds = oc. The H R F describes the way in which the instantaneous probability of 'death' for an individual changes with time. The function takes several shapes such as (monotone) increasing (IHR), decreasing (DHR), bathtub (BTHR) or other shapes. Models with I H R are used the most as they represent aging (wear out) in time. Information about the nature of the H R F is helpful in selecting the model. For detailed discussion of the HRF, see, for example, Lawless (1982). The expected value of the random variable T is known, in engineering, as the mean time to failure (MTTF), see, for example, Billinton and Allan (1983). A cdfF(t) is said to have an increasing hazard rate average (IHRA), i f - (1/t) log[R(t)] is increasing for t > 0. Reference may also be made to Shaked and Shanthikumar (1994) and

143


Chapter 33 in Johnson et al. (1995) for elaborate discussions on these reliability functions and their inter-relationships. A closely related concept is the mean residual lifetime (MRLT) which is defined, at time t, to be

m(t) = E [ T - t l T > t ] ,

t>O

F

1 (T - t ) f ( T ) d T R(t) oo ft+T 7 f0 exp [-- Jt r(x)dx]dT.

(2.6)

It can be shown that the M R L T is related to the RF by the following relation:

~S

ra(t) =

(2.7)

R(x)dx .

The RF is related to the M R L T by the relation

m(0)

F

p dx ]

R(t) = ~ t ( t ) - e x p [ - j 0 m~ßJ '

(2.8)

Furthermore, it can be shown that lim m(t) = }im

t~OO

(

- d td logf(t)]

)

(2.9)

The M R L T at time t, is a measure of the expected remaining life for an individual who survived up till time t. The above concepts extend naturally to more than one variable. For example, the multivariate survival function is defined by R(t_) = P[T1 >_ t l , . . . , Tn >_ tn],

ti > O, i = 1 , 2 , . . ,

n ,

where t = (tl, t2,..,tn). Johnson and Kotz (1975) (see also Kotz et al., 2000) defined a hazard rate vector r(t) by (2.10)

r(t) = (rl (t), .., r~ (t)),

where ô rj(t) = - ~ j ( l o g [ R j ( t ) ] ) ,

j= 1,2,..,n

.

Suppose that F(tlO ) represents a cdf of a random variable (vector) T given that O = 0 and that G(O) represents the cdf of the random variable (vector) O. The function H(t), defined by H(t) =-

Y

F(tlO)dG(O ) ,

oo

(2.11)

144


(which is the marginal cdf of T) was called by Fisher (1936) compound distribution of F and G. Teicher (1960) called H a mixtnre of F and G. F(tlO ) is known as the kernel and G the mixing distribution. If the entire mass of the corresponding measure of G is confined to a countable number of points 0 ~ , 0 2 , . . and the masses at Oj Ü = 1 , 2 , . . ) are G(@, then (2.11) takes the form (30

H(t) = Z F(t]Oj) G(0y) .

(2.12)

j=l

In this case H(t) is a countable mixture cdf. The noncentral X2 distribution is an example of a countable mixture of Poisson and central Z2 distributions. If the entire mass of the corresponding measure of G is confined to only a finite number of points 0 1 , 0 2 , . . , 0k, then (2.11) becomes a finite mixtnre of k components, whose cdf is given by k

H(t) = Z F(tlOy)G(Oy) .

(2.13)

j=l

To simplify notation, write pj

=_ G(Oj) and/~)(t) = F(t IOj), so that (2.12) becomes

(2 0, j = 1 , 2 , . . and ~pj j=l

=

o(oj) = 1 . j=l

Also, (2.13) becomes k

H(t) = ~ p j ~ ( t )

,

(2.1»)

j~l

where pj _> 0, j =

1 , 2 , . . , k and

k

~pj j=l

k

: ~

G(oj>:

1.

j=I

In (2.14) and (2.15), pj is known as the j t h mixing proportion and Fj(t) the jth eomponent in the mixture. In (2.15), k represents the number of components. It may be noticed that the choice Fj.(t) _= F(tlOj) restricts the cdfF(tlOj) for all values of j, to belong to the same family of distributions. However, forms (2.14) and (2.15) are written in the most general forms in which each of the cdf's Fy(t) could belong to a distinct family. The only requirement here is that, for any j, Fy(t) is a cdf.


145

Whether T is continuous or discrete, we shall write f(t]O) and h(t) to represent the conditional and marginal density (or mass) functions, respectively. If, in (2.11), G(O) is absolutely continuous, a pdf O(O) exists such that g(O) = G~(O) and if h(t) and f(t]O) are the p d f s corresponding to the cdfs H(t) and F(tlO), then from (2.11), we have

h(t) =

/?

f(t]O)g(O)dO .

(2.16)

O0

Similarly, corresponding expressions to (2.14) and (2.15) are given by OO

h(t) = Z p j J ) ( t )

(2.17)

j=l

and k

h(t) = ~pjfs.(t)

,

(2.18)

j--1

where J)(t) is the j t h component density (or mass) function corresponding to the cdf Fj(t). Barlow and Proschan (1981) defined the hazard transform of the mixture (2.11) as t/(u) = - log

F

e -"° dG(0) ,

OQ

where the vector u has elements uo, 0 < uo < oc, - o c < 0 < ec. They also showed that the hazard transform of a mixture is concave and that the mixture H(t), given by (2.11) is decreasing hazard rate D H R if each F(t I0) is D H R . It is decreasing hazard rate average D H R A if each F(t] O) is D H R A . They remarked that mixtures of I H R (IHRA) distributions are not necessarily I H R (IHRA). Ler

~n,,n = {F(tlO): t C IR", 0 C IR~} , be a family of cdfs and = { H : H(t) = f

JN 7

F(tlO)dG(O), G E N}

be a class of mixture distributions H generated by ~-~,m, where n and m a r e integers _> 1, IR~' is a Borel subset of Euclidean m-space IRm, F(t]O) is measurable in IR" × IR~' and .~ is the class of n-dimensional distributions G whose induced measures ga assign measure one to IR~. The class H of mixtures of ~-ù,m is called identifinble, according to Teicher (1961), if the mapping of N onto H is one-toone. Simply, G c N is said to be identifiable in the mixture H C ~ if a unique solution G of the integral in (2.11) can be found. Any H C ~ is termed a

146


countable (finite) mixture if the entire mass of the corresponding #c is confined to a countable (finite) set of points in IP,T. Identifiability of finite mixtures requires that k

k*

Ep~~I,I = EpT;(tl, j=i

v,,

j-1

implies that k = k*, pj = p* (j) and Fj = Fi*(/), for j = 1 , 2 , . . , k and some permutation i on 1 , 2 , . . . , k. Obviously, the identifiability question should be settled before one can meaningfully discuss the problem of estimating the mixing cdf G on the basis of observations from the mixture H. So estimation of the mixing distribution, testing hypotheses about G, etc., can be meaningfully discnssed only if the family of mixing distributions is identifiable. Several methods are used in estimating the RF of a mixture. For a given t, the RF is a function of the parameters. So, estimating the R F amounts to estimating a function of the parameters, in some methods. In others, estimating the RF does not involve a direct estimation of the parameters. The graphical method, the methods of moments, moment generating function, ML, Bayesian, minimum distance and the numerical decomposition of mixtures have been used. For details, see, for example, Titterington et al. (1985). The reliability function R(t) corresponding to a finite mixture of k components is given by k

R(t) = E p j R j ( t ) j=l

,

(2.19)

where R j ( t ) , j = 1, 2 , . . , k, is the RF corresponding to the j t h component in the mixture. According to the invariance principle, the MLE of a function of the parameters (RF or H R F for a given t) is the function of MLEs of the parameters. So, to obtain the MLEs of an RF or H R F , we simply replace the parameters by their MLEs. The problem then becomes finding the MLEs of the parameters of the mixture. For the finite mixture (2,18), with k components, the H R F is given by

r(t) - h(t) R(t)

(2.20)

'

where h(t) and R(t) are given, respectively, by (2.18) and (2.19). In this chapter, reliability, hazard and mean residual lifetime functions for some finite mixture models will be considered. It can be shown that in the case of only two components (i.e., k = 2), the H R F and M R L T of a mixture may be written in terms of the H R F s and MRLTs of the two components as follows:

r(t) = A(t)rl (t) + [1 - A(t)]r2(t) ,

(2.21)

m(t) = A(t)ml (t) + [1 - A(t)]m2(t) ,

(2,22)

ReIiability and hazard based on finite mixture models

147

where

A(t)

= p R 1 (t) /~gR1

B(t) = (1

- -

(t)

-~-

(1 - p)R2(t)] = [1 + B(t)] -1 ,

p)R2(t)/[pRl(t)]

(2.23)

.

So that

r'(t) : A(t)r' 1(t) + [1 -

A(t)]/2(t ) -

A(t)[1 - A(t)l[rl (t)

r2(t)] 2

-

(2.24)

m'(t) = A(t)m'~ (t) + I1 - A(t)]m'2(t ) - A(t)[1 - A(t)] [ml (t)

-

m2(t)]

2

.

(2.25) By observing that A(t) and 1 - A ( t ) assume values in the interval [0,1], for all t, it follows from (2.24) that if rj(t) < 0, for all t, j = 1,2, then r'(t) < 0, for all t. Therefore, a mixture with D H R components has DHR. However, if the components have IHRs their mixture need not have IHR. It may also be remarked that in the case of identical components (having identical parameters), the mixture reduces to the single component case. The same interpretation can be drawn for ra(t). Expressions (2.21), (2.22), (2.24) and (2.25) shall be used later. Since this chapter will be devoted to only finite mixture models, we present two groups of such models according to whether the components belong to the same family or to different families of distributions.

3. Models with components belonging to the same family Seven finite mixture models in which the components belong to the same family of distributions shall be introduced in this section.

3.1. Mixtures of normal components If, in (2.18), aä(t) is the N(]1:, a 2) density funetion, given, for j = 1 , . . ,k, by B(x) -

1 exp v~~crj

-2\

«: / - - O0 < X < O0,

--00 ;2/~'2 "

(3.27) (3.28)

From (3.28) and (3.29), the M R L T will eventually exceed the mean lifetime ,u of the mixed model whenever min(2/~2/21,2/~2/22) >/~ . Figures 3(a) and (b) show the graphs of pdf's and HRFs of two components of Inverse Gaussian and their mixture for some value of the vector of parameters

0 = (p» ~/1,/'/2, il'l, Ä2). 3.4. Mixtures of exponential components If, in (2.18), J)(t) is the exponential density, given for j = 1 , . . ,k, by fj(t) = ~ e x p { - t / c g } ,

t > 0, c9 > 0 ,

(3.29)

then, (2.18), represents a mixture of k exponential components. The exponential distribution has been the most commonly used distribution in reliability studies. Titterington et al. (1985) reported the use of mixtures of exponential components in medicine (renal xenon levels, conception times), geology (particle sizes), failure times (valve lifetimes, transmitter receiver lifetimes and reliability) and radioactive tracers. Mendenhall and Hader (1958) estimated the vector of parameters 0 -- (p, «1, c~2) of a mixture of two exponential components based on Type I censored data. Balakrishnan and Ambagaspitiya (1989) used a two-component mixture exponential as an alternative in evaluating some goodness-of-fit tests for exponentiality. Papadopoulos and Padgett (1986) estimated the parameters and reliability function of a mixture of two exponential compo-

158


(a) h(t) fl(t) f2(t)

0.8

0.6

0.4 •.

t '~

0.2

0.0 0

2

(b)

4

I

6

r

ù .-'"

1.2

r(t)

• -"

.........

,"

rl(t) r2(t)

0.8 -'1-

0.4

0.0

";

t /

.... ' ......................................

«

I

1

2

4

6

Fig. 3. (a) Density functions: Inverse Gaussian components and their mixture 0 = (0.5, 1.5, 2.5, 1.0, 17.0)• (b) HRFs: Inverse Gaussian components and their mixture 0 = (0.5, 1.5, 2.5, 1.0, 17.0).

nents based on right censored samples and using the Bayes method. AL-Hussaini (1999b) obtained the Bayesian prediction bounds for future observables based on Type I censored sample drawn from a finite mixture of two exponential components. Other studies were made by Rider (1961), Proschan (1963), Kleyle and Dahiya (1975), McClean (1986) and Willamain et al. (1992), Engelhardt (1995), among others. Chapter 19 of Johnson et al. (1994) and the book by Balakrishnan and Basu (1995) are devoted to exponential distributions, and they do include discussions on mixtures. Teicher (1963) showed that a finite mixture of k exponentiäl components is identifiable.

Reliability and hazard based onfinite mixture models

159

The reliability and hazard rate functions corresponding to a mixture of k exponential components are given by (2.19) and (2.20), respectively, where for j=l,...,k

Rj(t) = exp{-t/c~j}

(3.30)

and 1

r/(t) = - -

c9

.

(3.31)

For a finte mixture of two exponential components (k -- 2): (i) if cq = c~2 = c~, the mixture reduces to a single exponential component with a constant H R F c« (ii) if cq ¢ (E2, it follows from (3.32) that r}(t) = 0, for all values of t (j = 1,2). Therefore, from (2.24), r'(t) < 0, for all values of t, so that r(t) is decreasing on (0,oo). Case (il) is the (mixture) case which says that the mixture has a D H R eren though each component in the mixture has a constant hazard rate. Explanations for this result, considered as a paradox, were made by Barlow (1985), Rodrigues and Wechsler (1993) and more recently by Mi (1998).

3.5. Mixtures of Rayleigh components Il, in (2.18), ~(t) is the Rayleigh density, for j = 1 , . B(t) = 75exp -

,

t > o, ~j > 0 ,

,k, given by (3.32)

then, (2.18) represents a mixture of k Rayleigh components. The reliability function of a finite mixture of Rayleigh components is given by (2.19), where for

j= l,...,k

[ «)~]

Rj(t) = exp -

~

,

(3.33)

and the H R F 4 0 is given by (2.20), where ])(t) and Rj(t) are given for j = 1 , . . ,k, by (3.33) and (3.34), respectively. A mixture of two Rayleigh components is given by (2.18), where k = 2 and for j = 1,2, ]~(t) is given by (3.33). The three-dimensional vector of parameters 0 is denoted by 0 = (p, Ctl, az). Vodä (1976) discussed the moments method of estimation of the vector 0. This work is an extension of that by Krysicki (1963) who had earlier discussed the problem of estimation for a two-component mixture of Rayleigh distributions (see Chapter 18 of Johnson et al., 1994). It can be shown that the maximum

160


likelihood estimates of p, cq and c~2of the vector 0, may be obtained by solving the following system of three equations: /7

~~~(t,)

= 0,

- Y2(t,)]/h(t,)

i=1

~[bj(t,)fj(ti)]/h(ti) ] = O, j =

1,2

»

i=I

where h (t~) is given for k = 2 by (2.18) and fj(t~) by (3.33), j = i, 2 after indexing t by i and for j = 1,2, bj(ti) = 2 ( t ~ - c~~)/~3. The resulting estimates can then be used to obtain the MLEs of the RF and H R F for the mixed model. The H R F for a mixture of two Rayleigh components may be examined by using the fact that the H R F rj(t),j = 1,2 increases linearly. Then H R F for the mixed model increases (decreases) if (2.23) takes the form A (t)rll (t) + [1 - A (t)]r~ (t) - A (t)[1 - A(t)] [ra (t) - r2 (t)] 2 > (0,

aj, bj>O,

(3.38)

then, (2.18) represents a mixture of k Gompertz components. AL-Hussaini et al. (1999) showed that a finite mixture o f k Gompertz components is identifiable. For


(a)

I

10

h(t)

........

I

fl(t) I

163

I

,,'"

..

"',,

-õ

0.0

0.1

0.2

0.3

(b)

i s

120 ........ .......

tt

r(t) rl(t) r2(t)

t t s s/ t /

80

J t t I J "/r

40

0.0

t

t

0.1

0.2

0.3

Fig. 5. (a) Density functions: Weibu11 components and their mixture 0 (0.2, 3.0, 5.0, 0.25, 0.2). (b) HRFs: Weibull components and their mixture 0=(0.2, 3.0, 5.0, 0.25, 0.2). TM

details on mixtures of Gompertz components, see, for example, Gordon (1990), Adham (1996) and AL-Hussaini et al. (1999). The reliability function of this model is given by (2.19), where for j = 1 , . . , k

Rj(t) = e x p { - ~ ( e ajt- 1)} and the HRF r(t) is given by (2.20), where for j = 1 , . . , k , fj(t) and given by (3.38) and (3.39), respectively.

(3.39) Rj(t) are

E. K. AL-Hussainiand IC. S. Sultan

164

The Gompertz distribution has been used as a growth model, among other uses. AL-Hussaini et al. (1999) surveyed some of the uses of the Gompertz distribution. A finite mixture of Gompertz components model may represent different proportions that die due to different causes. In the above reference, estimates of the parameters, RF and H R F are obtained under a mixture of two Gompertz components model based on Type I and Type II censored samples. The likelihood and Bayes methods have been used in the estimation. We shall write X ~ Gomp(a, b) to denote that X follows a Gompertz distribution with parameters a, b. Let X ~ Gomp(a, b) with pdf given by (3.38) and T = bX. Then T ~ Gomp(2) with pdf

f(t)=exp{2t-~(e~t-l)},

t>0,

2>0,

(3.40)

where 2 = a / b . It may be observed that a Gomp(2) is actually a Gomp(a -- 2, b = 1). The likelihood function based on Type I censored sample is given by n

L()o; t) = II[h(ti)] ô~[R(to)l~i ,

(3.41)

i=1

where h(t) and R(t) are given, respectively, by (2.18) and (2.19), 2 = (21,)L2), 6i is an indicator function, given by 6i=

{1, 0,

Tl < to, T/>t0

(3.42)

and to is a predetermined time. By taking the logarithm of L in (3.41), differentiating with respect to the respective parameters, setting to zero and solving, we can obtain the MLEs )-1, )~2 of 21,2» The M L E of R(t), for a given t, is then given by

kML =-pRi(t) + (1 - p)k2(t) ,

(3.43)

where, for j = l, 2,

Rj(t) = e x p - ~ ( e x p ~ i t - 1 )

.

(3.44)

Similarly, the likelihood function based on Type II censored sample is given by

L*(Z; t) - ( ~ !

h(t«/)

[e(t(r/)] n-r ,

(3.4»)

where h(-) and R(.) are as given by (2.18) and (2.19) and t(i) are the ordered times for/=l,...,r. For a given t, the M L E R~L(t ) is then given by (3.43), after replacing ~by *, where R~(t) is given by (3.44) after replacing ^ by , 2"1 and 2*2 are the MLEs based on Type I1 censoring.

165


It is well known that the posterior density function is proportional to the product of the prior density and likelihood function. So, if ~z*(_2lt),fr(_2) and LF denote the posterior, prior and likelihood functions, respectively, then (3.46)

=*(_21_t) c< rc(_2)LF(_2,_t) .

The Bayes estimate of a function 0(.2) of the vector of parameters 4 is given by

B - E[0(4)IZ = _tl = £ 0(.2)~*(_2l_t)dL = [ 4(4>(_2)LF(_2;_t)d_2///~(_2)LF(_2;_t)d_2 . ùle

(3.47)

/Je

AL-Hussaini et al. (1999) used the following density function as a prior: :z(.2) = 3:1721exPl

41-17142__-1.]72 ' 4 j > 1, (Tj > 0 ) , j = l , 2

.

(3.48)

For a given t, if we set q5(4) = R(_t), given by (3.32), we can then obtain the Bayes estimate RB(t) ofR(t) based on Type I censoring if LF(_2, t) in (3.46) is replaced by L(4, t_) given by (3.41) and the Bayes estimate R~(t) of R(t) based on Type II censoring if LF(.2, t) is replaced by L*(4, t) given by (3.45). The ratio of the integrals in (3.47) may be approximated by using a form due to Lindley (1980) which reduces, in the case of two parameters, to the form

B = 0(2_) + S/2 + Pl&2 + P2S21 -~- [h;ov12 q- h~1c12 -- h~2c21 q- h;3v21]/2 , (21,42), S = ~~:1 •j2« Ovaij. Vor i,j = 1,2, Oij = ~2~/(~4i~,~j), 0, 1,2, 3, t/+ v = 3, aij = (i,j)th element in the matrix Z, where Z = -IJ(.2)1-1, J(.2) = [e2e/(e4,a#)], e = log LF(.2;_0 Vor / ¢ j, Sij : 4ilYii -- ~Sjaji , l)ij = (4iŒii ~- 4j(7ij)Œii, cij = 30iaii~Tij q- ~)j(aii(7jj ~- 2a2), where qSf = ~qS/ô4i and p~ = ~p/~4i, p = log n(.2), n(_2) is as given by (3.48). In all cases, the functions used in the approximation form are to be evaluated at the MLE of 4. Figures 6(a) and (b) show the graphs of pdf's and H R F s of two Gopertz components and their mixture for some value of the vector of parameters where _2

hùv aù+~e/(e470~), t/, v #

0 = (p, a l , a 2 , b l , b 2 ) .

4. Models with components belonging to different families

Different types of failure could lead to finite mixture models with components belonging to different families of distributions. In this section, four such models will be presented.

166


(a) h(t)

........

fl(t)

.......

.

."" ss«

,, -" ....

-, "',

ItpS

•• % %

0.2

0.1

0.3

0.4

(b) 70

l

r(t) tl(t) r2(t)

........ .......

l #

l

7

l

# tl l

50

ll

Il_

Il s

-1"

i s #1

30

llJ

10

0.1

0.2

0.3

0.4

Fig. 6. (a) Density functions: Gompertz components and their mixture 0=(0.2, 10.0, 15.0, 0.1, 0.2). (b) HRFs: Gompertz components and their mixture 0=(0.2, 10.0, 15.0, 0.1, 0.2). 4.]. Normal-exponential m i x e d model

Suppose fl(t)

1 = ~exp{-~

1

(~)

2

"~,

J -ool ,

(4.10)

(4.11)

, (4.12)


171

The M R L T corresponding to the mixed model is given by (2.22) (see Abd-E1Hakim, 1985), where ml (t) is given by (3.27) and

m2(t) = c~exp[(t/c)C]F(1 + 1/«; {t/~} c) - t , with £(/~;z) = f ~ x~-12e-Xdx. Figures 8(a) and (b) show the graphs of pdfs and H R F s of the Inverse Gaussian-Weibull mixed model and the components for some value of the vector of parameters 0 = (p,/~, 2, «, c).

(a)

~

~

~

1.5

" ,' .

-,

-

ù'"

,

',

I

- ........

h(t) fl(t) f2(t)

I

".,

1.0 13-

•

0.5

0.0

.

t

i

h

0.1

0.6

1.1

1.6

1.1

1.6

(b) 3

i ----.L.--

i ' ' "

«

r(t) rl(t)

,'

Un," "1-

0.1

0.6 t

Fig. 8. (a) Density functions: I G - W components and their mixture 0 = (0.7, 1.0, 1.0, 1.0, 4.0). (b) HRFs: I G - W components and their mixture 0 = (0.7, 1.0, 1.0, 1.0, 4.0).

172


4.3. Weibull-exponential mixed model Suppose that, in (2.18), k = 2, fl (t) and f2(t) are given, respectively, by fl(t)=

im)

r(~)~l

expl-

(~)',~ ~

,

t>0,

c~>0, c > 0

(4.13)

and

1

B(t) = ~ e x p [ - t / f l ] , P

t > 0, fi > 0 .

(4.14)

The resulting model, h(t) = Pfl (t) + (1 - p ) f 2 ( t ) , is known as the Weibull-exponential mixed model; see, for example, Ashour (1987). The reliability function of this model is given by R(t) = pR1 (t) + (1 -p)R2 (t), where R 1(t) = e x p [ - ( t / @ C]

(4.15)

R2(t) = expl-t/il] .

(4.16)

and

The H R F corresponding to the mixed mode1 is given by (2.21), where rs(t) = («/c~)(t/~) «-1

and

r2(t) = 1/fi .

Ashour (1985) obtained the MLEs of 0 = (p, a,c, fi) and the approximate asymptotic variance-covariance matrix, based on Type 1 censored data from the mixed Weibull-exponential model. Ashour (1987) considered the same problem by using Bayesian technique and compared the Bayesian estimates, based on a conjugate prior, with the MLEs. It is well known that the Weibull distribution has decreasing, constant or increasing hazard rate according as c < 1, c = 1 or c > 1, respectively. It then follows that the Weibull--exponential mixed model has D F R if c < 1, where c~¢ fi. Figures 9(a) and (b) show the graphs of pdfs and HRFs of the Weibull-exponential mixed model and the components for some value of the vector of parameters 0 = (p, ~, c, fi). 4.4. Modified semi-Poisson model In studying the distribution of time gäp in road traffic, Buckley (1962) suggested the use of a 'displaced' exponential (or two-parameter exponential), with density function

1

I

t-c~

to be a good fit for low-medium flow of traffic.

173


(a)

t

{

ù

1.5

I

I

I

"..

........ .......

h(t) I fl(t) f2(t)

1.0 I%

0.5

0.0 0.1

(b)

0.6

1.6

i

i

2.1

2.6

10.00 r(t) rl(t) r2(t)

........ .......

7.75

u_

1.1

5.50

n-

3.25

1.00 .

,."

0.1

i

i

i

i

0.6

1.1

1.6

2.1

2.6

Fig. 9. (a) Density functions: Weibull-exponential components and their mixture 0 = (0.7, 1.0, 4.0, 0.5). (b) HRFs: Weibull-exponential components and their mixture 0 = (0.7, 1.0, 4.0, 0.5),

A mixture of two dispaced exponentials, with density function, given by (2.18) with k = 2, where for j = 1,2, B(t)=~exp

],

t>~j, ej>0,

flj>O

(4.18)

was suggested to fit a situation in which the flow was made up of two types of vehicle, normal (free-flowing) and heavy (restrained). Ashton (1971) modified the 'semi-Poisson' model, proposed by Buckley (1962), by assuming the 'zone of emptiness' to follow the gamma instead of the normal distribution suggested by Buckley. In this model, a distance of length Z behind


174

each vehicle, which other vehicles never enter, defines the 'zone of emptiness'. A proportion p of vehicles stay at exactly distance Z from the preceding vehicle. The remaining proportion 1 - p follow at a distance Y behind the preceding vehicle. The random variable Ig is assumed to follow the exponential density with parameter 2. Ashton's modified semi-Poisson model is thus a finite mixture of two components with density function f ( t ) = Pf1 (t) + (1 - P)f2 (t), where 1

-1

fl(t) - F(-:)fi ¢t~7

exp(-t/fl),

t > 0, 7 > 0, /~ > 0

(4.19)

and B(t)

--

2 exp(-2t) F(~) I(?;t//~)(l+2fl) '~,

t>0,

2>0,

7>0,

/~>0 , (4.20)

where I(a; b) is the incomplete gamma function defined by

I(a, b) =

wa-le -w dw .

The component fl (t) is the density function of the random variable Z. Ashton (1971) derived f2(t) by first observing that its cdf is given by

F2(0 = P[Y _< tlY > z] - »[z _< t] ' P[Y< >y Z]

(4.21)

where Z has the pdf (4.19) and Y is assumed to have the exponential pdf

9(Y)=2exp(-2Y),

y>0,

2>0

.

By differentiating both sides of (4.21) with respect to t, we obtain

f2(t) = ddt {P[Z < Y < t]}/P[Y > Z]

d {fo,foyfl (z)9(y)dz dy }/f~ f o~fl (z)9(y)dy dz

= dt

.

(4.22)

The form off2(t), given by (4.20), then follows by substituting fl(z) and 9(Y) in this expression. For details, see Ashton (1971). The name modified semi-Poisson model is probably due to the fact that/'2 (t), in (4.20), is related to a partial Poisson sum through the incomplete gamma integral. In order to avoid the possibility of zero gaps, t is replaced by t - c~, in both of (4.19) and (4.20), to insure a minimum gap of c« A special case of f (t) is obtained by putting 7 = 1, which corresponds to an exponential zone of emptiness. In this case, f~(t)=~exp

-

,

t>~,

c~>0, / ~ > 0

(4.23)


175

and B(t) =2(1 + 2]~)exp { - 2 ( t -

~)}

x {1-exp[-(~)l},

t>cq c~> 0, f i > 0 .

(4.24)

Ashton's study was actually based on a finite mixture model of the two components with pdf's given by (4.23) and (4.24). Such a model was compared with the displaced exponential model given by (4.17) and mixture of displaced exponential model given by (4.18), which were suggested by Buckley (1962).

(a)

P

0.6

I

I

--

h(t) fl(t)

........ .

.

.

.

.

.

.

f2(t)

0.4

0.2

0.0 2

J

L

3

4

5

t

(b)1 5 ........ .......

r(t) rl(t) r2(t)

1.0

-----

. . . . . . . . . . . . . . . . . . . . . . . . . . .

0.5

jt t

0.0

i i

i

t

2

3

4

t

Fig. 10. (a) Density functions: modified semi-Poisson components and their mixture 0 = (0.5, 1.0, 1.5, 1.0). (b) HRFs: modified semi-Poisson components and their mixture 0 = (0.5, 1.0, 1.5, 1.0).


176

Table 1 Reliabilities and hazard rates corresponding to the jth component in a mixtue of k components belonging to the same family jth Component

Rj(t)

rj(t)

1 - gO[uj(t)] b

( Ju21//1 \(,!j~l/2ex ~) p{--g j f/~

(1) Normal: N(#j, a2) (2) Lognormal: A(#j, a2) (3) Inverse Gaussian: IG(@, 2j)

2% -exp{Tf}g)Ivj(t)l (4) Weibull: W(cj, ~j) (5) Exponential (~j) (6) Rayleigh («j) (7) Gompertz (aj, bi)

Cb[uj(t)l.

-exp{~} (b[vj(t)])

exp[-(t/~j) Cj]

(cj/c~j)(t/c~j)~~-1

exp[- ~ (e«,t - 1)]

bj exp(ajt)

a 05(.) is the pdf of standard normal distribution ~(.). b L/j(t) ~ (Äj/t)I/2(t/[2j - l) and vj(t) =

(2)/t)l/2(t/@ ÷ 1).

If a mixture of the two densities fl(t) and f2(t) given in (4.23) and (4.24), respectively, could be used in a lifetime situation, then the reliability functions of the components are given by Rl(t)=exp[-

( ~ - ~ ) l,

t>c~

and

It then follows that the H R F s of the components are given, for t > c~, by 1 r 1(t) = ~ = a c o n s t a n t hazard rate,

r2(t)=)~{1--exp [ - - ( ~ ) ] } / { 1 - - ( 1

+2~B2~)exp E - - ( ~ ß - ~ ) ] } "

Figures 10(a) and (b) show the graphs of pdfs and H R F s of the proposed mixed model and the components for some value of the vector of parameters 0 = (p, ~,/~, ~). 5. Summary and remarks

In this chapter, finite mixture models have been presented with discussion on their reliability, hazard rate functions and mean residual lifetimes. The models have been classified into two groups depending on whether the components of the

177


Table 2 Reliabilities corresponding to each component in a mixture of two components belonging to two different families Two-component mixture model

Forms of R1 (t) and R2 (t)

(1) I G - W

Rl(t)

L-t~)

1-(j~[@)l/2(~_ l)l - e~~F [2,1/2,tip+

1)1

Ra (t) = expE-(t/d)o] (2) Normal-exponential

R1 (t) = 1 - ¢ ( ~ ) R2 (t) = exp[-t/c~]

(3) Weibull--exponential

R1 (t) = exp[-(t/c~)cl R2 (t) - - expl- t/e]

(4) Modified semi-Poisson

R, (t) = e x p { - ( t ~ ) } , t > R2(t) = {1 + 2 f i - 2 f i e x p [ - ( ~ ) ] } e x p [ - ) o ( t - ~ ) ] , t >c~

mixture belong to the same family or to two different families of distributions. The first group covers seven models of finite mixtures with components belonging to the normal, lognormal, Inverse Gaussian, exponential, Rayleigh, Weibull and Gompertz families. The second group covers four models of two components each comprising the normal-exponential, Inverse Gaussian-Weibull, Weibullexponential and modified semi-Poisson models. Applications of the underlying mixed models as well as their components have been pointed out. Estimation of the RFs and HRFs of such models, by using different methods of estimation, have also been discussed. In addition, the MRLT of some of these models have also been examined. The RFs and HRFs of the components are tabulated in Tables 1 and 2 for these two groups of models.

References Abd-E1-Hakim, N. S. (1985). Finite mixture of the Inverse Gaussian and Weibull distributions. Ph.D. thesis, University of Assiut, Assiut, Egypt. Adham, S. A. (1996). On finite mixture of two-component Gompertz model. M.Sc. thesis, King Abdulaziz University, Jeddah, Saudi Arabia. Ahmad, K. E. (1982). Mixtures of Inverse Gaussian distributions. Ph.D. thesis, University of Assiut, Assiut, Egypt. Ahmad, K. E. (1988). Identifiability of finite mixtures nsing a new transform. Ann. Inst. Stat. Math. 40, 261 265. Ahmad, K. E. and E. K. AL-Hussaini (1982). Remarks on the non-identifiability of finite mixtures. Ann. Inst. Star. Math. 34, 543 544. Aitchison, J. and J. A. C. Brown (1966). The Lognormal Distribution with Special Reference to its Uses in Economics. Cambridge University Press, London. AL-Hussaini, E. K. (1999a). Predicting observables from a general class of distributions. J. Stat. Plann. Infl 79, 79-91. AL-Hussaini, E. K. (1999b). Bayesian prediction under a mixture of two exponential components model based on type 1 censoring. J. Appl. Stat. Sci. 8, 173-185.

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N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

Mixtures and Monotonicity of Failure Rate Functions

Moshe Shaked and Fabio Spizzichino

In this chapter we have selected to review some aspects of mixed distributions that have been of interest in our own research. First we describe some results that involve the preservation (or the lack of it) of monotonicity of the hazard rate functions of mixed distributions. Next we cover some recent advances in the study of the limiting behavior of the hazard rate function of a mixed distribution. Then some discussion on a multivariate ultimately negative aging property of multivariate exchangeable mixtures is given. Finally we describe some results on the distance of a univariate scale mixture from its parent distribution.

1. Introduction

The analysis of qualitative properties of the failure rate function of an individual, who is a member of a population of (at least apparently) 'similar' individuals, is fairly important in many fields where waiting times, attached to single individuals, are the object of study (by 'similarity' of individuals we may understand that the unidimensional distributions of single lifetimes are all identical). In the field of reliability theory, such properties are crucial when deciding whether a burn-in procedure or some type of preventive replacement politics are to be contemplated. For instance, a burn-in procedure may be recommendable when the failure rate function is decreasing, at least in some right neighborhood of the origin (this is the case, for example, when the hazard rate function has a bathtub shape). For reasons that we shall see in part later in this chapter, such a behavior can manifest itself in the case of mixed distributions. This is the reason why, in the field of reliability theory, the theme of burn-in is strictly tied with that of mixed distributions, and with the related orte of heterogeneous populations (see, for example, Block and Savits, 1997 and references cited therein). In this chapter we review some works which model observed lifetimes as observations from a mixed distribution. This is a common assumption in survival analysis and in reliability theory when it is believed that the sampled population is heterogeneous. Because of the multitude of applications of mixed distributions in 185

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M. Shaked and F. Spizzichino

these areas of research, m a n y advances have been made during the last 30 years in studying such distributions and their properties. We obviously cannot cover all the research that has been done with mixed distributions in reliability theory. In this chapter we have selected to review some aspects of mixed distributions that have been of interest in our own research. In Section 2 we describe some results that involve the preservation (or the lack of it) of monotonicity of the hazard rate functions of mixed distributions. In Section 3 we cover some recent advances in the study of the limiting behavior of the hazard rate function of a mixed distribution. Some discussion on a multivariate ultimately negative aging property of multivariate exchangeable mixtures is given in Section 4. Finally, in Section 5 we describe some results on the distance of a univariate scale mixture from its parent distribution.

2. Monotonicity of the hazard rate of a mixture

Imagine yourself in a queue for a server, say a bank teller, or an airline representative at an airport. The customer ahead of you has just stepped forward to the teller, and you are the next in line. That is, once the service given to the person ahead of you is done, then it is your turn. You probably feel optimistic - in just a short while it is going to be your turn, and then your waiting in the line will be over. However, after a short while you notice that the customer ahead of you is still being served... A little more time passes, and this customer is still being served... You start to have that foreboding feeling that the customer ahead of you "is one of those whose service takes for ever" (indeed, in most such situations you later realize that it has taken quite a while until you were finally served). This is the case even when each customer requires an exponential service time (that is, the service time has the lack-of-memory property) with a mean that m a y vary from one customer to another. W h a t has been described above is an illustration of the mathematical fact which states that a mixture of distributions with decreasing failure rate ( D F R ) is D F R (see, for example, Theorem 4.7 in page 103 of Barlow and Proschan, 1975). In particular this is the case even when the mixed distributions are exponential (that is, having constant failure rates). As can be seen from the above description, 'the more you wait, the more likely you are to further wait'. In other words, your waiting time seems to be D F R . The fact that sometimes a mixture of distributions, that do not have decreasing failure rate functions, m a y be D F R has fascinated m a n y researchers (see some references below). In particular, a mixture of distinct exponential distributions (each of which, of course, has a constant failure rate function) must have a strictly decreasing failure rate function. This fact may look at first paradoxical: If each component in a mixture has a constant failure rate function, that is, the lack-ofmemory property, should not the mixed distribution have the same lack-ofm e m o r y property, rather than the strict D F R property? The answer is that this need not be the case. When one observes an item with a distribution that is a

Mixtures and monotonicity of failure rate functions

187

mixture of exponential distributions, one actually gains information (rather than having a lack of memory) as time passes, as long as the item does not fail. The information that has been gained by time t > 0, if the item is still alive then, indicates that the actual realization of the mixing distribution is relatively more likely to be an exponential distribution with a large mean than with a small mean, as compared to the initial information at time 0 when no survival has been observed yet. In fact, survival at time t' > t indicates even more strongly that the actual realization of the mixing distribution has a large mean. To see it more clearly, imagine yourself again in a queue as described earlier. It is reasonable to assume that different customers have different service distributions. When you observe a customer (for instance, the one ahead of you) that has used the server for some length of time, and that is still being served, you are justified in supposing that this particular customer is likely to be one with a 'larger than average' mean, and therefore you are justified in expecting the rest of the service time to be stochastically larger than the service of an 'average customer'. In fact, the longer the customer ahead of you uses the server, the more convinced you are that he has a large mean, and the more likely it is that your remaining waiting time is going to be large. That is, your waiting time has the D F R property, even if the service time of each customer is exponential (with different means for different customers). The above discussion can be formalized and extended as follows. Let O be the parameter determining the survival function of the observed waiting time T, that is, let us set P{T>t[O=0}-F(tl0),

t>O

.

Also, let M be a mixing distribution (that is, a probability distribution for O), and let S be the set of possible values of O. Then the survival function of T is of the form

ffM(t) ~ PM{T > t} = fsff(tlO)dM(O),

t>0

(2.1)

For simplicity let us assume that Æ(t]0) > 0 for all t > 0, 0 E S. I f w e also assume that Æ(.I 0) is absolutely continuous for all 0 E L, then we obtain that /~M is absolutely continuous as well, and its associated failure rate function is given by

IM(t)

liml

4- At) - FM(t)

1 .fs[F(t + AtlO)-F(tlO)]dM(O) - ~ - At+oAt fsF(tlO)dM(O) =1' 1 1 f F(t + AtlO)-- F(tlO) )~,õ At fsF(t]O-)dM(O) Js ~=="~-~ltjo) F(tkO)dM(O) . l

= im-

Notice at this point that, whenever interchange of the limit operation with integration is allowed, we can write


188 g

ru(t) = ffs r(tlO)dM(OIt),

t>0 ,

(2.2)

where r(.lO) denotes the failure rate function corresponding to Æ(.lO), and where

f(tlO)dM(O) dM(OIt ) = f ~ )

.

(2.3)

For T > 0, the mixing distribution M(OJt) defined by (2.3) can be given two, slightly different, interpretations, according to the interpretation given to M.

Frequentist interpretation. We deal with a multitude of similar individuals with associated waiting times /'1, T2,.., each T~ with its own parameter 0i. If M is interpreted as the distribution of the values of Oi's, then M(-It ) is the distribution of the subpopulation of the 'residual' Oi's, attached to the only individuals for which T« > t.

Subjective interpretation. We consider a single, observable, lifetime T and an associated non-observable parameter O with the assigned set of conditional survival functions/~(t[0) for T given O = 0. I f M is the 'a priori' distribution for O, then M(.[t) is, by Bayes formula, the 'a posteriori' distribution of O, after observing the survival data T > t (see Spizzichino, 1992). In this view we can write

r~4(t) = E[r(tlO)lT > tl,

t _> 0 .

(2.4)

It is clear from (2.2) and (2.3) that qualitative properties of r(tlO ) and M(O) provide relevant information about the monotonicity behavior of rM(t). However the analysis is greatly complicated by the fact that the mixing distribution M(.It) actually varies with t; thus general results are very hard to achieve. It can be of interest, in this marter, to take into account possible properties of stochastic monotonicity of T in O. Notice that those properties can reflect on stochastic monotonicity of O in T and then on monotonicity properties of the distribution M(.]t) as a function of t. Assume, for instance, that S is a subset of the real line and that [T[O = 0] is increasing in 0 with respect to the hazard rate ordering; that is, ff(tlO')/F(t]O ) is increasing in t whenever 0 _< 01 (see, for example, (1.B.2) in Shaked and Shanthikumar, 1994), of, in other words, that F(tl0 ) is a totally positive (TP2) function of t and 0 (see, for example, Karlin, 1968). It is easily seen from (2.3) that, in such a case, dM(O)) taust be TP2 as well; that is, for 0 < t < t I we have [ol:r >

t]

--~lr [OlT > t'l ,

where _ t] _<s~ [olr > t'] ,

Mixtures and monotonicily of failure rate functions

189

where 0. Rajarshi and Rajarshi (1988) and Gupta and Akman (1995) have described some mixture models that yield bathtub failure rate functions. The reader may have picked up the impression that every mixture of distributions with increasing failure rate (IFR) has a failure rate function that strictly decreases over some region of [0, oc). This is, of course, not true. For example, Lynch (1999) has noted that a convolution of two I F R distributions is IFR, and such a convolution can also be viewed as a mixture of I F R distributions. There


190

are many other examples of mixtures of IFR distributions that are IFR. This motivated Lynch (1999) to look for general conditions under which the IFR property is preserved under mixtures. Lynch (1999) proved two results of the kind described above. Consider a survival function/~M of the form (2.1).

FM(t) = f Æ(tlO)dM(O),

«> 0

where _P(-]0) is a survival function for each 0, and M is some mixing distribution. The first result of Lynch (1999), which follows at once from Prekopa's Theorem, says that if: (i) F(t[0) is logconcave in (t, 0), and if, (ii) the mixing distribution M(O) has a logconcave density ra(O), then ÆM is logconcave, that is, it has the I F R property. The second result (Theorem 2.1 of Lynch, 1999) says that if: (i') il(tl0) is logconcave in (t, 0) and is non-decreasing in 0 for each t, and if, (ii') the mixing distribution M(O) is IFR, then, again, FM has the I F R property. It is remarkable to note that the result that a convolution of IFR distributions is IFR, is a special case of Theorem 2.1 of Lynch (1999). Some further studies of mixtures in the context of monotonicity (or lack of monotonicity) of associated hazard rate functions can be found in Gurland and Sethuraman (1995). They obtained necessary and sufficient conditions for a mixture of two IFR distributions to be DFR. They also studied the special case of (2.1) where

FM(t) =

f #(t)dM(O), ~>_0 ,

(2.5)

that is, when the distributions that are being mixed have proportional hazard rate functions. Again, when F is IFR, they gave necessary and sumcient conditions for FM to be DFR, when M is either a two-point distribution or a continuous distribution. Gupta and Gupta (1996) also studied mixtures of the form (2.5). More explicitly, they studied (and also gave a few references to other stndies of) some particular mixtures, such as gamma mixtures of Weibull distributions, and inverse gamma mixtures of Weibull distributions. It is worthwhile to note that the assumption of proportional hazards (2.5) is very common in survival analysis because of its mathematical tractability. An interesting study of mixtnres of piecewise exponential distributions, with applications to biology and to mining disaster data, can be found in Zelterman et al. (1994). We have mentioned above that a mixture of, say, m subpopulations, each with a distinct constant failure rate, taust have an overall decreasing failure rate. This


191

is the case when each of the m subpopulations is modeled independently of each other. That is, when the m subpopulations are assumed not to interact with each other. In other words, this is the case when a 'death' of an individual means a removal of that individual from the respective subpopulation and also from the aggregate population. But in m a n y demographic studies, a removal of an individual from one subpopulation may mean its transfer to another subpopulation. For example, a labor force exit may mean an entry of the individual to another subpopulation. Similarly, changes in marital status mean transitions among subpopulations. Rogers (1992a, b) modeled such transitions and showed that then, even when each subpopulation has a constant failure rate, the aggregate failure rate need not decrease. A more general analysis of aging properties for heterogeneous population, where individuals can change subpopulations along time, is au interesting field and, at the best of our knowledge, one still open for research.

3. Limiting behavior of the hazard rate of a mixture

As at the beginning of Section 2, imagine yourself again at the head of queue, waiting for the current customer to finish his service - at that time your wait in the queue will end. Suppose now, however, that there are only two kinds of customers: Either 'fast' ones that on the average require a short service time, or 'slow' ones that require on the average a lengthy service time. When it is also assumed that customers require exponential service times, then the above statement means that the service time of the 'fast' customers has a high (constant) failure rate, whereas the service time of the 'slow' customers has a low (constant) failure rate. Now, the longer it takes for the customer ahead of you to conclude his service, the more convinced you become that that customer is one of the 'slow' ones (this statement can be translated into a formal mathematical statement and then easily proven). After a while you become quite convinced that the current hazard rate of the service time of the customer ahead of you is actually the hazard rate that is associated with the 'slow' customers. That is, a long observed 'survival' t of the customer ahead of you at the server leads you to believe, with a probability that tends to 1 as t ---+ec (this can be proven mathematically), that the current hazard rate, at time t, is the low one which corresponds to 'slow' customers. What has been described just above is an illustration of the mathematical fact which states that the hazard rate function of a mixture of distributions tends to follow in the long run the lowest failure rate function (if such one exists) among the failure rate functions of the distributions that are being mixed. In particular, if the lowest failure rate function is increasing (decreasing), then the failure rate function of the mixture will tend to have the same property in the limit (that is, for large t). A formal mathematical statement that conveys the above intuitively clear idea may not be easy to state and prove. However, in some special cases (for example,

192


when there are only two distributions that are being mixed, or when the distributions that are being mixed are exponential, or have decreasing failure rate functions) this formal mathematics has been worked out in the literature. We informally describe below some of the advances in this area of research. Consider a mixture of the form given in (2.1). Ler S denote the support of M, and for any 0 E S let r(.lO ) denote the hazard rate function associated with F(.IO ) (that is, r(tl0 ) = -(~/~t)log F(tlO), provided it exists). Suppose that the limits a(O) = lim~__+~r(tlO ) exist for all 0 E S, and ler ~ = inf{a(0) : 0 C S} (that is, c~is the limiting hazard rate corresponding to the 'strongest' distribution (in the limit) among the distributions that are being mixed). Let rM denote the hazard rate function associated with FM (that is, rM(t)=-(d/dt)log/~M(t), provided it exists); see also (2.4). In a study of burn-in and mixed populations, Block et al. (1993) showed, under some regularity conditions, that rM(t) --+ c~ as t --+ oc. In other words, in the limit, the failure rate of the mixture follows the smallest limiting failure rate; see also Block and Savits (1997). Mi (1999) further refined this observation by finding conditions under which the above intuitive conclusion also holds when it is formally restated by means of limiting conditional survival probabilities, or by means of limiting mean residual life functions. Rather than looking just at the limiting value of the hazard rate functions, but also looking at the monotonicity properties in the limit, Block and Joe (1997) refined the results Block et al. (1993) in a different direction. Most of their results apply to mixtures of two lifetime distributions. Ler us denote by rl and r2 the hazard rate functions of the two distributions that are being mixed, and as before, let rM denote the hazard rate function of the mixed distribution. Suppose that one component is stronger in the limit than the other one in some sense; without loss of generality, the stronger component may be selected to be the second one. Block and Joe (1997) postulated it formally as the requirement that rl (t) _> r2(t) for any large enough t. Block and Joe (1997) then found conditions under which, for example, (rM(t)/r2(t)) ,[ 1 as t ---+ee. They also extended these results to more general mixtures. Gurland and Sethuraman (1995) have shown that mixing exponential distributions with various other distributions lead to failure rates that are ultimately decreasing (that is, that are decreasing on regions of the form (to, oc) for some to < oc). Block and Joe have noted that these results of Gurland and Sethuraman (1995) are really special cases of their results in Block and Savits (1997). Let us give here a word of warning. It should be pointed out that the study of the limiting behavior of a hazard rate function is of importance only in situations where long survivals are crucial and are emphasized. The reader taust have noted already that if the mixed distribution FM is the distribution of the life of an item, then the probability is often small that the age of the item ever reaches the time region in which the above mathematical results apply. Most items will expire a considerable time before that. Thus, in situations where the interest centers around the 'average' lifetime of the item (such as in the study of light bulbs in a theater hall), one should avoid trying to apply the above mathematical results, unless one has a convincing reason for doing that.


193

4. Negative aging properties of multivariate exehangeable mixtures In this section we review some results of Spizzichino and Torrisi (2001) regarding exchangeable mixtures and their negative aging properties. The starting point of Spizzichino and Torrisi (2001) is that if Tl, T2, • •, T~ are non-negative independent and identically distributed random variables, then each is D F R if, and only if, for any choice of ages h , . . . , ti-1, t l + l , . . . , tj-1, tj+l,.., tn (i < j), and any ~ > 0, the following implication holds: ti < tj =~ P { Ti > ti + ~ID,~} ~ tj +

~lDn}

,

(4.1)

where Dn = {Tl > tl,T2 > t2, ..,Tn > t~} . The validity of the implication (4.1) provides a notion of negative aging also for the more general case when the lifetimes Tl, T 2 , . . , T~ are merely assumed to be exchangeable rather than independent and identically distributed. This notion can be viewed as a natural analog of the D F R property for the case of exchangeable lifetimes. For more discussion on this point of view, see Bassan and Spizzichino (1999). Sometimes (4.1) may hold only for large enough ti's. That is, suppose that the exchangeable random lifetimes Tl, T 2 , . . , Tn satisfy the implication ti < t] ~ P{Ti > ti + z[Dn} < P{T] > tj+ z[Dù},

whenever tz > t, l = 1 , 2 , . . , n

(4.2)

for some t _> 0. Then we say that Tl, T 2 , . . , Tn have the multivariate ultimately negative aging property with respect to ~. In particular, when n = 1, then (4.2) reduces to the assumption that P{T1 > t~ +v]T~ > tl} is non-decreasing in h E It, ec) for all -c > 0; in such a case we say that T1 (of its survival function) is ultimately D F R with respect to t. Spizzichino and Torrisi (2001) studied the following model of multivariate exchangeable mi×tures. Consider a population of individuals 1 , 2 , . . , n . Let (Zi, Ti) be a pair of random variables associated with individual i. We think of as an observable lifetime of individual i, and of Zi (taking on values in ~ for all i) as an unobservable quantity (often called frailty) which determines the distribution of Tl.. Since Zi is random, the marginal distribution of T~ is a mixture. Suppose that the individuals are 'similar' in the sense that given ~ = z, the distribution of Tl. is determined by the value z, independently of i. That is, suppose that there exists a family of univariate survival functions {G(.Iz), z E ~ } such that given (Zl, Z 2 , . . , Z,) = (zl, z2, .., zn), the conditional survival function of Tl. is G(.Izi). We also suppose that given Z = z, the random variables Tl, T 2 , . . , Tn are independent. We assume below that Z1, Z 2 , . . , Z , , are exchangeable and this implies that T~, T 2 , . . , Tn are also exchangeable.

194


Let us suppose that G(.[z) is ultimately D F R with respect to sorne t for all z E ~q. Furthermore, let us suppose that

ä(tlz') - -

G(tlz)

in non-increasing in t _> ~ whenever z < z' .

Spizzichino and Torrisi (2001) have shown that then T ] , T 2 , . . , T n have the multivariate ultimately negative aging property with respect to t. Some variations of this result, which give conditions under which each marginal distribution is ultimately DFR, are also described in Spizzichino and Torrisi (2001).

5. S o m e results on univariate scale m i x t u r e s

In this section we specialize (2. l) to mixtures of the following form: FM(t) =

/0 ~

F(Ot)dM(O),

(5.1)

t > 0 ,

where M is a distribution function of a positive random variable O. In terms of random variables we have (=st denotes equality in law) T =st Y/O

.~

where Y and O are independent random variables with distribution functions F and M, respectively, and T is distributed according to F~t. If we denote by p the failure rate function associated with the F in (5.1), then the failure rate functions, conditional on O = 0, are given by r(tlO) = Op(Ot),

t >_ 0

(5.2)

and, by specializing the formula (2.4), we obtain rM(t) = E [ O p ( O t ) l T > t I ,

(5.3)

t >_ 0 .

Both, the proportional hazard model described in (2.5), and the scale model described in (5.1), are obtained by somehow incorporating a random effect to a baseline survival function F. These two models are then parallel, in a sense, but, in general, are different one from another. The two models coincide when the baseline survival function is exponential. For a same initial mixing distribution M, it can be of interest to compare r~ in (5.3) with the expression of rM associated with the model given by (2.5). In this respect, using the symbols r ~ ) and r ~ ) for the expression in (5.2) and for the corresponding expression associated with model (2.5), respectively, we can write

r(~/ -r(~ ~ -----Cov(«/E(O,

p ( O t ) ) l r > t] + p(t)Æ(»)[OI r > t]

+Æ(s)[oIr > d " e(s)[p(ot)l

r

>

t],

t >_ o ,


195

where the suffixes (S) and (P) remind us that the conditional distribution of O given T > t is computed according to the model in (5.2) or to the one in (2.5), respectively. In particular,

EIPt[OIT > t]

E(s)[OIT > t]

L O[Æ(t)]° dM(O) = L[p(t)]OdM(O ) ,

t »_ 0 ,

fL O[Æ(Ot)] dM(O) -- LE~(0t)] d M ( 0 ) '

t > 0

Note that, trivially, Cov (s) [(69, p(Ot))lT > tl) is positive or negative according to whether p is an increasing or a decreasing function, independently of the value of t. General results concerning the comparison among r(~ ), r~ ) and p are hard to achieve. Several aspects of the scale change models are examined in Anderson and Louis (1995), and, for some particular cases, the differences among the scale change model, the proportional hazard model and the baseline model without heterogeneity are illustrated graphically. It is a common practice to approximate the reliability of an item with distribution function FM, given in (5.1), at time t (that is, FM(t)) by ~e(t), if orte has a reason to believe that EO or EO -1 is nearly equal to 1. Thus it is of interest to obtain bounds on suptlF(t ) --FM(t)I = suptlÆ(t) --ÆM(t)I. Few authors have obtained various general results which give useful bounds in some interesting special cases. Here we will not describe these general results, but we will describe the resulting bounds in some special cases. Suppose that FM is a scale mixture of Weibull distribution functions with shape parameter c~ > 0; that is, suppose that in (5.1) we have

F(t)

= 1 - exp{-~t~},

t > 0

for some fixed 2 > 0. Shaked (1981) has shown that in this case sup IF(t) - FM(t)I < E[O ~ - 11 , t

provided the expectation exists. Note that equality holds if, and only if, the mixture FM is pure; that is, if, and only if, O is degenerate at 1. Take ~ =- 1 to obtain that if FM is a scale mixture of exponential distribution functions then sup IN(t) - FM(t)[ 0 for some c5 > 0}, and the survival function Æ(x) = 1 - F ( x - ) is given by {'(x) = I x g x ( 1 - - VF{Xr})] exp{--H«(x)} ,

x 0 ((5) also holds with 'x < b' replaced by 'x c R' provided we define exp{-oc} to be equal to zero).

COROLLARY 2.6. I f --oC < C~< OC and the restriction of F to (-o% c~) is continuous (i.e. if vF is continuous or non-atomic on (--oc, @), then Æ(x) = exp{-H(x)}

for all x E ( - e t , c~) ,

where H(x) = VF((--oGx]) and we define exp{-oc} = 0.

202

M. Asadi and D. N. Shanbhag

We have taken the definitions and results appearing above from Kotz and Shanbhag (1980). Specialized versions or variants of these have appeared in Cox (1961, 1972), Jacod (1975) and other places. The following are some of the standard definitions appearing in the literature, but with modifications to take into account recent improvements to these suggested by Kotz and Shanbhag (1980) and Keilson and Sumita (1982). Most of the literature still restricts itself, while addressing the issues linked with the definitions involved, to the distributions concentrated on R+ with infinite right extremities, and the cited references are among those that have made an effort to study larger classes of distributions with members that are not necessarily concentrated on R+. DEFI~TION 2.7. The random variable X is said to be smaller than the random variable Y in the usual stochastic order (st), denoted by X _<stY (or by Y _>stX) if B(x)- 2c(X)

for almost all [PF + Pc]x Œ R ,

where PF and Pc are the measures determined on R by F and G, respectively. The usual stochastic ordering is being used in many areas of statistics and applied probability. For some standard references on this, we refer the reader to Marshall and Olkin (1979), Ross (1983) and Shaked and Shanthikumar (1994). DEFINITION 2.9. Let X and Y be two random variables with X + and Iz+ integrable and the mean residual (mrl) functions mF and inc, respectively. Then X is said to be smaller than Y in the mrl order, denoted by X -<mrl Y (or by 1z ~mrl X), if mF(x) 0}, and it implies that bF _< bG. Also, note that if X - 0} and VF({x E R: G(x) = 0}) = 0. Moreover, if VF({X e R : G(x) • 0}) ~---0 and { X n : rl = 1 , 2 , . . } and {Yn: n = 1 , 2 , . . } are

sequences of randorn variables converging in distribution to X and Y, respectively, such that Yn ~hm Xn, then Y ~hm X. PRoov. We have from the representation (5) =exp

Z

(log(1 - v«{xr}) - l o g ( 1 - va{xt}))

l ~H~F~ ~x~ - 4 a l ( x ) ) ~,

x < min{bF, bG} ,

(8)

J where b F and bc are the right extremities of F and G, respectively, D (F) is the set of discontinuity points of vF lying in ( - o o , x ) and H(~F)(x) = VF,«((--OO,X]) with VF,« as the non-atomic part of VF, and D~(c) and H~(G)(x) are defined similarly for G. If y _< x < min{bF, bG} and Y (~) >/>(y) .

(9)

G(x) - G(y) to see this note that (in obvious notation)

/>(x)

/>(y)

f

O(x) = G - ~ e x p ~

OO

x,.6(D(F)uD!G))\(D~F)uD~,G)) k=l +

v~,«((y,x]) - ~~'«((Y'~J)/ '

assertion Y- bc ve((bG, oo)) > vc((b~, oo)), and bF < ba ~ for an a smaller than bF and sufficiently close to bF, VF((a,bF]) > vG((a, bF]).) Because of the left continuity of/> and G we have then the "only if" part of the first assertion in view of (9). To prove the "if" part of the first assertion, note that under the given condition, The

be = ba and the exponent of (8) is an increasing leit continuous function on (--0o, bF) with limit as x ~ - o c to be zero. Consequently it follows that for each x E (--oC, bF), VF{X} O ,

(11)

giving fob~-X O(x + t)dt < för-x fi(x + t)dt ~(x) P(x) ' (12) implies Y ~mrl X.

x c ( - ~ , ò~) .

(12) []

Singh and Vijayasree (1991), using a counter example, showed that the mrl ordering is not closed under the formation of k-out-of-n systems. A comparison of random sums based on the mrl ordering is studied by Pellerey (1993), while Shaked and Shanthikumar (1994) showed that under some conditions the mrl ordering is preserved under the operation of taking convolution. Shaked and Shanthikumar (1991) also proved that under the condition that the ratio of the mrl's of X and Y is increasing, the hr ordering and the mrl ordering are equivalent. The following theorem now shows that the last result of Shaked and Shanthikumar (1991) stated above remains valid even when the assumption that random variables are non-negative and absolutely continuous is dropped.

206


THEOREM 3.6. Let Xi, i = 1,2 be two random variables with X~+, i = 1,2, integrable and mean residual life functions mi, i = 1,2, respectively. Let bFh, i = 1,2, be the

right extremities of the distribution funetions of Xi, i = 1,2, respectively. Suppose that ml(x)/m2(x) increases for x<min{bF~,bF2}. Then )(1 ~mrlÄr2 implies Xl ~hmX2 • PROOF. The assertion of the theorem can be proved as follows. As observed earlier, X1 --<mrlX2 implies that bF~ is less than or equal to b&. The increasing nature of ml (x)/m2(x) for x < min{b_bF2 and hence we have that bF, = bF2. On the other hand, we have ff2(X) __ tal(X) fx/P2(t)dt B(x) m2(x) f/~~(t)dt'

x < min{bFh, be2} .

Under the assumptions of the theorem, in view of what we have observed immediately after Definition 2.9 and the fact that bF~ = bF2, we have the right-hand side of the last equality, and hence its left-hand side, to be increasing on {x E R:/~1 ( x ) ) 0}, and in obvious notation, VF2({x E R : P1 (x) = 0}) = 0. By Theorem 3.2, we have then X1 ~hm X2, and the theorem is proved. []

4. Some closure properties of the hazard measure ordering In this section, we extend some of the known closure properties of the hr orclering involving either sums of independent random variables, or order statistics relative to a random sample, to the corresponding properties of the hm ordering. However, before doing so, we give the following two important results of auxiliary natnre which tell us that under appropriate assumptions, the hm ordering implies the st ordering of certain conditional distributions. THEOREM 4.1. Let Y ~hm AT and Z be a continuous random variable independent

of X and Y such that P{X > Z} > 0 (and hence also such that P{Y >_Z} > 0). Then (XlX ~ Z) ~st ( Y I r

_> z)

(13)

PROOF. Let us denote by F, G and H, respectively, the df's of X, Y and Z. We can then see that (13) is equivalent to

B,oo) H(y)dF(y) fR H(y)dG(y) - /

Jfx,oo)

H(y)dG(y) / H(y)dF(y) >_0 for all x ¢ R . dR

We can see that (13) is equivalent to the condition that

(14)

Hazard measureand mean residuallife orderings:A unifiedapproach

207

B,~) H(y)dF(y) £~,x) H(y)dG(y) - J~,~) H(y)dG(y) x~

H(y)dF(y)>O

a(- OO~X)

for a l l x C R ,

(15)

which, in turn, is then seen to be equivalent, in view of Fubini's theorem, to ùfRmin{P(x),P(z) } dH(z) >.f-oo,x](G(z) - G(x))dH(z)

-

]£ min{ G(x), G(z)} dH(z) __][-oo~](Æ(z)- F(x))dH(z) >_0 for all x c R. (16)

As the inequality in (16) is met trivially when G(x) = 0, it is clear that to have (13), it is sufficient if we show that for each x with G(x) > 0 and hence F(x) > 0,

~(x)O(x){~min{1Æ(z)

-fRmin{1

(G_-(z)

1) dH(z)

\~- 1)dH(z) } > 0 G-(z)-'~dH(z) f(_~,x](~(z)

' C(x)J

(17)

-

In view of Theorem 3.2, we have that (17) holds for each x with F(x), G(x) > O. Hence, we have the theorem. [] COROLLARY4.2. Let Y ~hm X and Z be a eontinuous random variable independent

of X and Y. Then, for all z with P{X + Z >_z} > 0 (and henee P{ Y + Z >_z} > 0), we have

(xlx+z >z) >_st(:qy + z _>z)

(18)

PROOF. The result follows on applying the theorem with z - Z in place of Z with z arbitrary. [] COROLLARY 4.3. If the assumptions of Corollary 4.2 are met with distribution of Z as absolutely continuous having an interval support and an increasing hazard function, then

Y + Z 1, x,, is a point such that Ü(xn) = 1/(n + 1) and the hazard rate atxn isyn, then one can take Xn such that P{X~ > x} = f •(x) I. F(xn)e -y'(x-x')

if x _< x~, i f x _> x,, .)

We have then X,, _ 0 and n -- 1,2,.... Suppose Y* is the random variable distributed as Y and independent of {Am}. As Y* has an absolutely continuous distribution with interval support and increasing hazard rate on the support, Corollary 4.3 implies that Xù + Y* ~hm Xn -c Y* 4- t for each t > 0 and n = 1 , 2 , . . . As {X,* + Y*} converges in distribution to X + Y and X 4- Y has an absolutely continuous distribution, the stability theorem for hazard measures given, for example, in Kotz and Shanbhag (1980) implies then that for every Borel subset B of ( - o c , b), where b is the right extremity of the distribution of X 4 - Y, vH(B)>_vH(B-t)

for a l l t > 0

,

(21)

where H is the distribution function of X 4- Y (with vH obviously as the corresponding hazard measure). It is obvious that the distribution of X 4- Y has its support to be an interval and density (i.e. some version of it) to be continuous. In view of this, (21) implies that the theorem holds. [] COROLLARY 4.7. Let (Xi, Y/), i = 1 , 2 , . . , m , be &dependent random vectors such that Yi _~hm)(/, i : 1 , 2 , . . m. [je Xi and Yi, i : 1 , 2 , . . , m, have absolutely continuous distributions with interval supports and increasing hazard rates on respective supports, then m

m

i=l

i-1

(Also Theorem 4.6 implies that the distributions of ~im~xi and ~im=l Yi have interval supports with increasing hazard rates on respective supports.) PROOF. We shall obtain the result by induction. Assume that it is valid when m = k, where k is a fixed positive integer. Then, if we define (Xk+x, Y~+I) to be a


210

random vector independent of (X,., Y~), i = 1 , 2 , . . , k and distributed as 0(1, Y1), we have by Corollary 4.3 and Theorem 4.6, Yii

q- Yk+l __~hm

Xi

47 Yk+l

i

+Xk+,

i=1

- 0}. Taking a hint from this, if G is a d f on R, we can define the reverse hazard measure relative to G as the measure v~ on R such that for every Borel set B

where Pc is the measure determined by G, on R. Note that for every Borel set B, ~b(B) = ~ ~ ( - B )

,

where H is the d f given by H(x) = 1 - G((-x)-),

x ER

and vLr is the hazard measure relative to H. Implications of this to our study are self-evident. The following three results (i.e. two theorems and a corollary) follow via shorter proofs in view of Theorem 3.2. However, as their direct proofs have some interesting features, we have decided to produce them here.

Hazard measure and mean residual life orderings." A unißed approaeh

211

THEOREM 4.11. Let (Xi, Yi), i = 1 , 2 , . . , m be independent two-component random vectors such that Xi __ ~i=1Y(i), It is easy to see that x -4 y implies both x -<w Y and x _<w y.

Some comparison results of the reliabiHtyfunctions of some coherent systems

217

The concept of node criticality introduced by Boland et al. (1989) is central to the following developments. The following notation will be used. Suppose x E {0, 1 }~, i.e. each one of the n components o f x is either 0 or 1. We use (li, 0j, x (ij)) E {0, 1} ~ to denote a vector w h o s e / t h c o m p o n e n t is 1, j t h c o m p o n e n t is 0, and the remaining components are the same as in x, vector (0i, l j, x (i~)) etc. will be defined similarly. DEFINITION 3 (Boland et al., 1989). In a coherent system said to be more eritical than node j for 4, denoted by i > j, ~b(0i, l j , x (ij)) for all x (ij) and strict inequality holds for at ~)(li,Oj, x (ij)) = ~(Oi, lj,x(iJ)),VX ~ {0,1} n, then no«es i and permutation equivalent, denoted by i ~=j.

(S, 4), hode i is if 4(li, Oj,x (U)) >_ least one x (ij). I f j are said to be

TnEOREM 1. Consider a coherent system (S, O) of m nodes 1 , 2 , . . , m. Suppose that l C 2 ~=... ~:n. Let a, b Œ~n (n < m) and q C ~m_n, where ~m-n = {(Œ1, .., C~m-n) : 0 < C~k< 1,k = 1 , 2 , . . ,m - n}. If l n a -< lnb, then h(a,q) _< h(b,q), and the equality holds if and only if a = b. Here h(a, q) is the reliability of (S, 4) when the components assigned to nodes { 1 , 2 , . . , n} and {n + 1 , . . , m} have reliability a and q, respectively. PnOOF. Since 1 c 2 ~ . • • & n from Boland et al. (1989) we know that the reliability function h(p, q) is permutation invariant in p ~ Nn. Thus, we can assume ak = a(k) and bk = b(k), 1 < k < n without loss of generality. Further, since l n a - < lnb, so in a can be derived from ln b by successive applications of a finite n u m b e r of T-transforms (see, e.g., Marshall and Olkin, 1979). We can assume that there are indexes 1 < i < j < n such that

lnai = 2 lnbi + (1 - 2) lnbj ,

(2)

lnaj = 2 lnbj + (1 - ,~) lnbi ,

(3)

for a constant 2 ~ (0, 1), and lnak = lnb~,Vk ¢ i,j. That is, l n a is derived from lnb by one T-transform. F r o m (2) and (3) we obtain ;~ (1-4) ai : b i bj , (4) a j : @ b } 1-2)

(5)

Hence aiaj = bibj. Moreover, it can be shown that ai 4- aj < bi + bi. In fact, the inequality

implies

i.e., ai -~- aj ~ bi ~- bj .

218

J. Mi

This further gives (6)

ai + aj - aia j < bi + bj - bibj .

Since aiaj = bibj yields either ai ph. This time we have ~7=~ lnp[i] > n lnp, or ~ i ~ l lnp?]/n >_p. Hence it follows that: ~~=1 lnp[~] > k -

Ein=l

lnp[~] >_ P n

V1 < k < n

and so k

k

Elnp[i]>-kp=Zlnp~l i=1

V1 < k < n .

i=1 st

That is, lnp* ~w lnp and thus Sn(p*) < Sn(p) by our Corollary 1. Theorem 1 (il) of n Ma (1997) can be shown by noticing that I~i=l qi ~_ qn where qi z 1 - p i and q - 1 - p , implies (lnq,... ,lnq) -~w (lnql,... ,lnqn). For the convenience of application we can convert our theorems to be stated in terms of lifetime of coherent system. Let 321,.. ,X~ be independent lifetimes of n KI~ • . . , K~, respectively. Denote components following distributions K = (Kl,... ,K~) and 1et z(K) be the lifetime of the aforementioned coherent system (S, ~). COROLLARY 3. Let (S, ~b) be a coherent system with m nodes and 1 o= 2 o= ... o= n. Suppose that one set of components has independent lifetimes T l , . . . , Tn with

220

J. Mi

distribution functions F1, .. ,Fn and W1, W%..., Wm-n with distribution function H~, ..,Hm_n, and another set has lifetimes S1, ..,Sn which are independent of W1, W2, .., Wù~-n and follow the distributions G1, .., Gn. I f

( l n ~ ] ( t ) , . . , l n F n ( t ) ) -% ( l n G l ( t ) , . . , l n G n ( t ) )

V0 < t < x ,

(7)

wh«r« Fi(t) ~ l - F i ( t ) and Gi(t ) ~- I - Gi(t)» then P ( z ( F , H ) > t) < P(z(G, H) > t), V 0 < t < x, and "c(F,H) is the Iif«time of (S, (o) when lif«tim«s of components assigned to node i(1 O, then st

-c(F, H) _< r(G, H) . REMARK 1. Suppose that (S, Ó) is a coherent system and has a modular decomposition {(A~,z1), (A2, z2)} where A 1 has n nodes and A2 has m - n nodes. Denote the nodes in A1 by 1 , 2 , . . , n and assume that they are permutation equivalent. Then the aforementioned results can be applied to this system. For the definitions of module, modular decomposition see Barlow and Proschan (1981). REMARK 2. For any a = ( a l , . . •, an), b = ( b i , . . . , bh) with ai > O, bi > 0 1 < i < n the majorization a-< b does not imply ln a - % ln b. (Actually, ln a-<w ln b implies a-<wb) However, a -1 -~Wb-a does imply l n a - % l n b , where a - 1 ( l / a 1 , . . , l/an) and b - l _= ( I / b i , . . . , 1/bn). Also, the weak submajorization n H in a -<w In b must imply ~i=1 a~ _< ~i=1 b~, V c~> 0. These results are shown in Marhsall and Olkin (1979).

3. Applications In this section the results developed previously will be used for different applications. For the simplicity of notation, we will state the results for k-out-of-n system. Slight modification can easily give the similar results for coherent system (S, ~b) with rn nodes of which n are permutation equivalent 1 ~ 2 ~ ..- =~ n. In these examples the conclusions will be expressed in terms of lifetime of the system. We will use the same notation such as F, G etc. defined in Section 2. Also, whenever the lifetimes 7 1 , . . , Tn are involved they are assumed to be independent. THEOREM 3. Let F/(t) have failure rate function and cumulative failure rate function ri(t) and Ri(t), 1 t ) , (ii) Suppose rl(t)

V0_ 0, then ,(F) < z(G). PROOF. For any given t we have lnfii(t) = - R i ( t ) and lnGi(t) = -Ai(t) by the

definition of cumulative failure rate function. Hence, from (R~(t),.. ,Rn(t)) - t) < P(r(G) > t),V0 < t < x. Now consider (il). From the assumption in this oase and the definition of weak supermajorization we have ~ / k l ri(t) > ~ik=l 2i(t),V 1 < k < n, and V0 < t < x. These inequalities immediately imply ~~=lRi(t)=~,~=lfóri(s)ds>~~_l Jó}~i(s)ds = Ek=l Ai(t), V l < Ic < n, and V0 < t < x. Therefore, from (i) we obtain (ii). [] EXAMPLE 2. Suppose F/ is exponential with failure rate ri and Gi is exponential with failure rate fli(1 < i < n). From Theorem 3 we see that if @ 1 , . . , rn) _<w ( 2 1 , . . ,fl~), then ~(F) < r(G). For instance, (3.5,4,7) -< (1,5,8) so the lifetimes of series, parallel and 2-out-of-3 systems consisting of components with exponential distributions exp(3.5),exp(4),exp(7) is stochastically smaller than their counterparts from exp(1),exp(5),exp(8) components even though r2 = 4 < 5 = 22 and r3 = 7 < 8 = 2» The following result generalizes Corollary 2.7 of Pledger and Proschan (1971). COROLLARY (Proportional failure rate). Let r(t) be a failure rate function and L = (21,..,)~~) and I1= (th,...,rl~) be two vectors with positive eomponents.

Suppose that Fi, Gi have failure rate functions 2ir(t) and tlir(t), respectively. I f st z «~ n, then ~(F) «_ ~(c).

PROOV. It is true since (flq,...,~n) _.~w (/'/1,''',/~n)

implies ()qr(t),..,2~r(t)) _~w O h r ( t ) , . . ,~,r(t)). follows.

Hence,

the

desired result []

EXAMPLE 3. Let F~-(t) = F/(t; cti, )~i), Gi(t) =- Gi(t; fii, ~i), 1 «1 . . . . . c~n -z ~ > fi ~ fil . . . . . fin > 0. If

J, Mi

222

( 2 1 , . . ,2n) _<w (r/l,..., t/,), then P(v(F) > t) < P(r(G) > t),Vt > max{1/2i, 1 _ (2it) ~, 1 /9. Then, same as in the proof of Lemma 1, we conclude that ((21t)~,.., (2~t) ~) _<w ((21t)/~,.., (2j)/~),Vt _> max{1/2i, 1 t) _ t),Vt >_ max{1/2~, 1 < i < n}. It is easy to see that in this example if we further assume the shape parameters st c~ =/~ c (0, 1], then the above result can be strengthened as z(F)_< z(G). In st particular, if c~ =-/~ C (0, 1] and 2~ . . . . . 2~ =- f/= ~~~_~ tli/n, then z(F) < ~(G) since ( 0 , . . , 0n) -< ( q l , . . . , %) always holds. In reliability theory, the mean residual life (MRL) function is another important quantity since there is a one-to-one correspondence between the set of lifetime distributions and the set o f M R L functions, In the following, for convenience we assume all the involved M R L functions are continuous without repeat. Out next result will use M R L function to compare the lifetimes of k-out-of-n systems. THEOREM 4. Let the associated M R L functions of Fi(t) and Gi(t) be i~i(t) and vi(t), respectively. Suppose that for a given x > 0 the following conditions are satisfied: 1° For each pair of indexes i < j the inequality #i(t) _< #j(t), V0 < t < x holds. 2 ° For each pair of indexes i < j the inequalities vi(t)< vj(t),V0 < t < x and vi(O)/vi(t) t), V0 < t < x. In particular, if the above condist

tions are true for all t > 0, then r(F) < z(G). PROOF. For any pair i < j from condition 2 ° we have -

ds . . . > _ _

~n(S

UI(S ) --

-

1 __

V0<s<x

Vn(S)

and thus by 4 ° further have for any 0 < t < x t

1

t

/0 ) l ~ d s > -

/0

t__1 #,-1 (s) ds + t

1

fo v ~ ) ds'

1

t

1 as >_ t

/o

t__1 ds + v, 1(s)

1

t

//1

ds,

v,~

1

t

v - ~ ds

V0 < t < x .

1

fo #~)ds+'"+fo #~)ds>-f v~ds+'"+ fo v~ ds" The inequalities imply t (-f0

1 ~ds,'",-f0

-<w -

t

1 ~

ds,...,-

d,)

The weak submajorizations (10) and (11) yield

(ll)

224

J. Mi

(ln#~(0)_

#l(t)

* 1

(lnV,(O)

V'w \

,ln#~(0)

t

1

f0 ~-7~ ds''' " ~--~ -- f0 ~t-7~ ds) '1

vl (t ) -- fO v-7-~ ds, " . , ln ~

'1

- foo v - ~ ds ) •

(12)

since the sequences at the right-hand sides of (10) and (11) are similarly ordered. Note that by the inversion formula we have Fi(t ) = #i(0) e-

fót~i~7 1 ds

and thus In Fi(t) = In #i(0) Bi(t ) - f0 t ~ 1

ds .

(13)

ds ,

(14)

Similarly, it is true that in äi(t) = in

-

Comparing (13) and (14) with (12), we conclude that (ln F l ( t ) , . . , l n

F,,(t)) -% (in ä l ( t ) , . . , l n

Gn(t))

V0 < t < x .

Therefore, the desired result follows from Corollary 2 of T h e o r e m 2.

[]

COROLLARY (Proportional MRL). Ler #(t) be a mean residual life function.

Suppose that the M R L function associated with F~(t) and Gi(t) are given as ~---,~ifl(t) and vi(t) = 7ifl(l), respe«tiv«ly. If k -1 _<w q-l, i.e.,

Bi(l)

st

the~ ~(v) < ~(c). PROOF. First of all, note that we can assume 21 _< 2 2 . . . < 2n and 71 -< 72 -< "'" -< 7, without loss of generality. Since #i(O)/#i(t) = vi(O)/vi(t) = #(O)/#(t), 1 < i < n, so conditions 1° and 2 ° of Theorem 4 are satisfied. Condition 3 ° is equivalent to ( 1 , . . , 1) _<w ( 1 , . . , 1) which is certainly true. Finally, condition 4 ° is equivalent to

st

which is what we assumed. Therefore, by T h e o r e m 4 we see that r(F)_
#i(t)/vi(t) increases in t E [0,x], 1 vi, 1 < i < n, t h e n u _~w v. F r o m this fact in case (a) we h a v e 3 °, a n d i n case (b) we h a v e 4 °. N o w i n case (c) the a s s u m e d c o n d i t i o n s i m p l y those r e q u i r e d in (a) a n d (b) a n d t h u s b o t h 3 ° a n d 4 ° are true. []

References Barlow, R. E. and F. Proschan (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD. Boland, P. J. and F. Proschan (1983). The reliability ofk out ofn systems. Ann. Probab. 11, 760 764. Boland, P. J., F. Proschan and Y. L. Tong (1989). Optimal arrangement of components via pairwise rearrangement. Naval Res. Logist. 36, 807-815. Glesser, L. (1975). On the distribution of the number of successes in independent trials. Arm. Probab. 3, 182-188. Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Arm. Math. Stat. 27, 713-721. Ma, C. (1997). A note on stochastic ordering of order statistics. J. Appl. Prob. 34, 785-789. Marshall, A. W. and I. Olkin (1979). Inequalities: Theory of Majorization and its Applications. Academic Press, New York. Mi, J. (2000). A Unified Way of Comparing the Reliability of Coherent Systems (submitted). Pledger, G. and F. Proschan (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Optimizing Methods in Statistics (Ed. J. S. Rustagi). Academic Press, New York.


12t _J

On the Reliability of Hierarchical Structures

Lev B. Klebanov and Gabor J. Szekely

In this paper a one-parameter family of limiting survival functions of hierarchical structures is associated to (almost) every reliability polynomial. These survival functions are generalizations of the exponential survival function (corresponding to the reliability polynomial of series structures). The problem of reconstruction of reliability polynomials from these limiting survival functions is also settled.

1. Introduction

Let N = N1 be an arbitrary network, and call it the first generation. Let us obtain the second generation N2 from N1 by replacing each component of N1 by N. Similarly, if we replace each component of N2 by N then we get the third generation N3, and so on. Suppose the components of N function independently and each component has the same probability p of proper functioning. Then the chance that N functions properly is a polynomial of p, called the reliability polynomial which will denote by h(p) (we follow the notation of Barlow and Proschan, 1975). There is a lot of information known about h(p) for monotone structures (we shall consider here only monotone structures in terms of Barlow and Proshan, 1975). It is easy to see that the nth function iteration of h is the reliability polynomial of An. Let us denote this nth iteration of h by h°n(p). The theory of iterative functions (see Kuczma et al., 1990) is a good tool to stndy these hierarchical networks. The limiting function F of the survival functions Fn of _~~ with linear normalization (as n tends to infinity) can be reduced to the solution of the equation

il(t) = h(F(zt)) for some z E (0, 1). A similar equation plays an important role in the theory of branching processes but there the role of h and of fi is played by the generating function or characteristic function of the corresponding distribution F (see Harris, 1963 or Athreya and Ney, 1972). This makes the solution of the above equation 227

L. B. Klebanov and G. J. Szekely

228

completely different. Equations of this form were introduced by Poincaré (1980) (see also Valiron, 1954). Poincaré's paper remains a very valuable source. Two of the properties of h(p) for monotone structures we are going to use are that h(p) is an increasing function o f p in the interval [0, 1] and it is S-shaped (for details see Barlow and Proschan, 1975). Thus h(p) can have at most one fixed point in the open interval (0, 1). It is natural to suppose that 0 and 1 are always fixed points: h(0) = 0 and h(1) = 1. Now consider a reliability polynomial h(p) and suppose that it has three fixed points in the closed interval [0, 1]: p = 0, p = 1 and p =Po. Let to and tl (h _< to) be arbitrary non-negative numbers and introduce the following survival function: /70(t) =

{

0, po, 1,

t > to, to--_ > t_> h, t
0) to a disc Itl < õ of the complex plain. In this case, using the fact that r ~ (0, 1), it is obvious t h a t f has an analytic continuation to the whole complex plane as an entire function. In the following theorem we suppose that the derivative h~(1) > 1. In the typical S-shaped form of h we have a (unique) solutionp0 o f p = h(p) in the open interval (0, 1). In this case instead of h~(1) > 1 we need h'(p0) > 1. For more details see Section 7.

On the reliability of hierarchical structures

229

THEOREM 2.1. Let h(p) be a polynomial in p such that."

(i) (ii)

h(1) -- 1 and h(p) > 0 in the interval (0, 1]," h'(p) > O for all p C (0, l) and h'(1) > 1.

Put r = 1/h'(1). Then for any a > 0 Eq. (2.1) has a solution f ( t ) analytic in the

disc ltl < c~, (~ > 0) and satisfying the conditions: f ( 0 ) = 1,

f'(0) = -a

.

(2.2)

This solution is' positive, monotone decreasing for t > O, unique in the class of functions satisfying (2.2), and analytic in the disc [t] < 6. REMARK. U n d e r the conditions of this theorem there is no solution po o f p = h(p) in the open interval (0, 1), otherwise (2.1) would imply that h(p) is identically equal to p, and thus h~(1) > 1 cannot hold. Using this observation it is easy to show that {01

F(t) =

- f(t)

for t < 0, for t _ > 0

(2.3)

is a cumulative distribution function. The p r o o f is the following. According to T h e o r e m 2.1 f ( 0 ) = 1, and f is positive, m o n o t o n e decreasing, therefore limt~o~f(t) = v exists. We only need to prove that v = 0. Since f satisfies Eq. (2.1) we can take the limit in b o t h sides of (2.1) as t ---+ oc. Thus we get v = h(v), that is v = 0 or v = 1. But the case v = 1 can be excluded since f ( 0 ) = 1 and f is strictly decreasing. EXAMPLE. If N is a series structure of k > 1 c o m p o n e n t s then h ( p ) = p h , h'(1) = k > 1 , r = l / k , and f ( t ) = e -at is the solution of (2.1) satisfying (2.2). PROOF. The p r o o f is divided into four parts. (i) Set f ( t ) = ~~=oant n, where a0 = 1 and al = - a . Then the coefficients an, (n _> 2) can be determined uniquely f r o m Eq. (2.1). T o see this, differentiate Eq. (2.1) n times with respect to t. F o r n = 1 we get

f ' ( t ) = h'(f(vt))f'(rt)'c

(2.4)

for n _> 2 we obtain n!

( f ' ( ' c t ) ~ mt (f(~)('ct))m"zn 1! J " " \ n! J

(2.5)

f(n)(t)=z---'ml!m~~..mn(h(s)oc(zt))\~"

where the sum ~ is taken over all non-negative integer m l , . . . , m n under condition ~ j = l jmj = n, and s = ml + ... + mù. Plugging t = 0 into (2.4) and (2.5) we see that the value of al can be arbitrary, and using (2.5) the values of an = f(n)(O)/n] for n >_ 2 are determined uniquely by a0 and al. (ii) The series ~n~0 aùt" converges in the disc Itl < 6, (6 > 0).

230

L . B. K l e b a n o v a n d G. J. S z e k e l y

Indeed, putting t = 0 into (2.5) and using h~(1) = 1/r we obtain f"(O) z" n! -- l - - - ~ n - l E m l ! . . .

an--

lm~

1! h/,)(1)a~,,

m°_, ""an-1

(2.6)

here m l + • .. + mù-i = s and the sum ~ is taken over all non-negative mj under the condition ~j~__-]j m j = n. We are going to show that [azJ _< AL l, l = 0, 1 , . . for some constants A > 0, eA < 1/~ and L > 0. Suppose that this inequality is proved for all l _< n - 1 where n is sufficiently large, and let us prove it for l = n. F r o m (2.6) we see that for n > 2 C 1 ih(~)(1)lASL" la"l -< 1 --7~-a Zml[...mn_l! --i-

C L~ 1 C -~ ~ml!...mn_l!

z" 0 for all t > O. I f f ( t 0 ) = 0 and f ( t ) > 0 for 0 < t < to then 0 = f ( t o ) = h(f(zto)) > 0 .

This contradiction proves (iv). The statement of T h e o r e m now follows from (i), (il), (iii), and (iv).


231

REMARK. It is obvious that if f is any solution of Eq. (2.1) satisfying the conditions of Theorem 2.1, then f has an analytic continuation from any interval of the form (0, 6) to the whole complex plane as an entire function.

3. The general solution of the main equation In this section we determine the general solution f of Eq. (2.1) without assuming that f is analytic. DEF~NrrIoN 3.1. Let fo be the solution of (2.1), satisfying the conditions

f(O)=O,

f'(O)=-I

and all other conditions of Theorem 2.1. In this case we say that fo is the standard solution of (2.1). TnEOREM 3.1. Suppose that the polynomial h satisfies all conditions of Theorem 2.1. Let fo be the standard solution of (2.1), and f be any other solution of (2.1)for t > O. Then there exists a In z-periodic function ~ such that

f ( t ) =fo(tq/(lnt)),

t> 0

(3.1)

and vice versa, for any lnz-periodic O the function (3.1) is a solution of (2.1)for t>0. PROOF. Since f0 is strictly monotone (decreasing), we can introduce the following function: ~(t) = ~fo'

ü'(t))

•

This means that f(t) =f0(t~(t))

.

From the equätions

fo(t) = hOCo(zt)) and

f ( t ) = h(f(rt)) we obtain

hOeo(zt(p(zt) ) ) = h(fo(zt~o(t) ) ) , which is equivalent to (2.1). But both h and f0 are monotone, therefore the previous relation is equivalent to

~o(~t) = ~o(t) .

L. B. Klebanovand G. J. Szekely

232

Put t = e ° and (p(e°) = 0(0). So, we have 0(0) = ~/(0 - in r) that is ~ is a in z-periodic function. The proof is complete.

[]

It can be interesting to find conditions that guarantee that the function ~ in (3.1) is constant. Our Theorem 2.1 shows that it is sufficient to suppose that f has analytic continuation from [0, 6). But this condition is too strong. PROPOSITION 3.1. Let Y be the class of functions given on R+, differentiable in an interval [0, 6) such that the derivative f ' satisfies the Hölder condition (3.2) below (with fixed parameters) in the underlying interval. Suppose further that the poIynomial h satisfies all conditions of Theorem 2.1 and maxzc[0,1] Ih'(z) l = h'(1). Then Eq. (2.1) has only one solution satisfying (2.1) within the class ~ . PROOF. Introduce the class Y, of all monotone decreasing functions from ~satisfying (3. l). All functions f in ~,~, satisfy the Hölder condition If(tl) -- f(t2)] _< KIq - t2lr ,

(3.2)

where K > 0 and r > 0 are universal constants for the class Y » Fix an arbitrary ~ (0, r) and introduce the distance:

«(fl,f2) = ~00°° ] f l ( t ~ ~ [ 2(t) l d t ,

(3.3)

in Æ,. The distance d is finite in o~, and with this distance Yl is a complete metric space. Consider now the operator T defined on ~ , by the relation

Tf(t) = h(f(rt)) .

(3.4)

We have

d(~,

/0 ~ Jh«l(~,/~~~~(~,//Id,

~/

_< h,(1)zl+~ f0 °~ Ifl(0t2+~ -f2(t)l dt =

~~d(A,f2)

.

Here we used the definition of ~ = hr(1). The inequality

d(Tfl, Tl2) 0 in interval [0, 1]; (ii) h'(z) >O for a l l z E (0,1) andh'(1) > 1. P u t ~ = 1/h'(1). Then for any a > 0 Eq. (2.1) has a solution f ( t ) , analytical in a circle It] < ~, (6 > 0) satisfying to the conditions." f ( 0 ) = 1,

S(0) = -a .

(6.1)

This solution is positive and monotone decreasing for t > 0 and unique within the class of analytic solutions in a circle It[ < ~ and satisfying (6.1). PROOF. The proof is very similar to that of Theorem 2.1. Copy (2.6) from the proof of Theorem 2.1 an

fn(O) _ C ~ 1.mn_l!h(,)(1)aT~ ... an-lm"-~ ~! 1 ---~~ 1 Z-~m~!..

(6.2)

for all n > 2. Since the series

j•h••(1) j!

( _ 1): ,z

converges for all z, we have that there exists an M > 0 and an t/ E (0, 1) such that

Ih«/(1)/j!j ~ M#


235

for all j = 1 , 2 , . . We are going to show that there also exists an L > 0 and an A > 0 such that ]aj] to we have F(t) < Æ(t0), and it is easy to see that h°n(Æ(t)) --+ to as n ~ oe. Therefore the limit distribution does not exist (the limit function is a constant, not a survival function). Introduce a new function

Fl(t) = F(to + t) . Obviously, ffl (0) : Po. T o study the asymptotic behavior of h °n (fr1 (t)) we need the usual linear normalization. Consider the asymptotic behavior of h°n(#l(Vt)) for fixed z Œ (0, 1). We can study this equation for the cases t < 0 and t > 0 separately, and thus we arrive at the situation we already considered in Section 2.

Referenees Athreya, K. B. and P. E. Ney (1972). Branching Processes. Springer, New York. Barlow, R. E. and F. Proschan (1975). Statistieal Theory of Reliability and Life Testing Probability Models. Holt, Rinehart and Winston. Eremenko, A. E. and M. Yu. Lyubich (1989). Dynamics of analytical transformations. Algebra Anal. 1(3), 1-70, (in Russian).

236

L. B. Klebanov and G. J. Szekely

Fatou, P. (1921). Sur les fonctions qui admettent plusiers théorémes de multiplication. C.R. Acad. Sci. 173, 571-573. Harris, T. E. (t963). The Theory ofBranching Processes. Springer, Berlin-Göttingen-Heidelberg. Julia, G. (1922). Mémoire sur la permntabilité des fractions rationnelles. An. de l' École Norm. Supér. 39, 131~15. Kuczma, M., Choczewski, B. and R. Ger (1990). Berative Function Equations. Cambridge University Press, Cambridge. Poincaré, H. (1890). Sur une classe nouvelle de transcendantes uniformes. J, Math. Pures Appl. 6, 313-365. Valiron, G. (1954). Fonctions Analytiques. Presses Universitaires De France, Paris.


1

lk ~j

Consecutive k-out-of-n Systems

N. A. Mokhlis

Notation

x/

state of component i (i.e. X~ = 0 or 1 according as component i fails or works) reliability of component i (Pi = Pr{X~ = 1}) failure probability of component i (i.e.

Pi

qi

qi = 1 - pi) 7

R~F(k; n; p l , . . . , p~)

R~o(k;n;pl,..

. ,pù)

R~F(k; n ; p )

R~o(k; n;p)

P~4

R,~F( (r,s); (n,m); Pl,1, . . ,Ph,m)

implies L (linear) or C (circular) reliability of a 7-consecutive k-out-of-n: F system with component reliabilities pi, i= 1,..,n reliability of a 7-consecutive k-out-n: G system with component reliabilities pi, i=l,...,n. reliability of a 7-consecutive k-out-of-n: F system with identical component reliabilities p, i.e. Pi = P for i = 1 , . . , n reliability of a 7-consecutive k-out-of-n: G system with identical component reliabilities p reliability of the lattice-system component located in /th row (circle) and jth column (ray) reliability of y-connected-(r, s)-out-of(n,m): F lattice system with component reliabilities pi,j; for i = 1 , . . n and j=

R~G((r, s); (n, m ) ; P l , 1 , . .

,Ph,m)

l,...,m

reliability of 7-connected-(r,s)-out-of(n, m): G lattice system with components

237

238

N. A. Mokhlis

reliabilities ])ic; for i = l , . . . , n and j= 1,..,m reliability of ~-connected-(r, s)-out-ofR~v((r, s); (n, m);p) (n,m): F lattice system with identical component reliabilities p reliability of 7-connected-(r, s)-out-ofR,yG((r, s); (n, m);p) (n,m): G lattice system with identical component reliabilities p RyF((r, «)-or-(s, r); (n, m);Pl,1,.., Ph,m) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n,m): F lattice system with component reliabilities pi,j; for i = 1 , . . , n andj=l,...,m R~G((r, s)-or-(s, r); (n, m);pl,1,.. ,Ph,m) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n,m): G lattice system with component reliabilities pi,j; for i=l,...,nandj=l,...,m R~v((r, s)-or-(«, r); (n, m);p) reliability of 7-connected-(r,s)-or-(s,r)out-of-(n, m): F lattice system with identical component reliabilities p reliability of pconnected-(r,s)-or-(s,r)R~G((r, «)-or-(s, r); (n, m);p) out-of-(n,m): G lattice system with identical component reliabilities p reliability of relayed unipolar consecutive Ru(k;n;pl,... ,pù) k-out-of-n: F system with component reliabilities pi, i = 1 , . . . , n reliability of relayed bipolar consecutive Rb(k;n;pl,... ,Ph) k-out-of-n: F system with component reliabilities Pi, i = 1 , . . , n reliability of relayed unipolar consecutive Ru(k; n;p) k-out-of-n: F system with identical component reliability p reliability of relayed bipolar consecutive Rb(k; n;p) k-out-of-n: F system with identical component reliability p. largest integer less than or equal to x

1. Introduction

The reliability of consecutive k-out-of-n systems has been studied by many researchers due to their practical importance. Consecutive k-out-of-n systems may be classified into two classes; consecutive k-out-of-n: F and consecutive k-out-of-n: G systems. Moreover, two cases could be considered; the one-dimensional case and two-dimensional case.

Consecutive k-out-of-n systems

239

1.1. The one-dimensional case (a) A eonseeutive k-out-of-n: F(G) system consists of an order sequence of n components, such that the system fails (operates) if and only if consecutive k (k _< n) components fail (operate). The system is called linear or circular depending on whether the components are arranged on a straight line or form a circle. The reliability of linear consecutive k-out-of-n: F system was first studied by Kontoleon (1980), but the name consecutive k-out-of-n: F was first considered by Chiang and Niu (1981). Derman et al. (1982) discussed the circular consecutive kout-of-n: F system for the first time and expressed its reliability in terms of the reliability of linear systems. (b) Relayed consecutive k-out-of-n: F systems was first discussed by Chiang and Chiang (1986). However, Hwang (1988) introduced the notion of relayed unipolar and bipolar consecutive k-out-of-n: F systems. For a relayed unipolar consecutive k-out-of-n: F system the components are arranged in a line such that the system fails if the first component (source) fails or at least k consecutive components fail. While for a relayed bipolar consecutive k-out-of-n: F system, the n components are arranged in a line such that the system fails if the first component (source) fails, last component (sink) fails, or at least k consecutive components fail. The one-dimensional consecutive k-out-of-n systems are of great need in out daylife. Chiang and Niu (1981) and Hwang (1988) indicated their relevance to some telecommunication, oil pipeline and microwave systems. Bollinger and Salvia (1982), and Fu (1986a) remarked that such systems frequently arise in the design of electronic integrated circuits. Kuo et al. (1990) gave an example of railway station. Example 1.1.1 (Telecommunication system given by Chiang and Niu, 1981). A telecommunication system with n relay stations (either satellites or ground stations) numbered from 1 to n. Suppose that a signal emitted from station 1 can be received by stations 2, 3 , . . , k + 1 together, and a signal emitted from station 2 can be received by stations 3, 4 , . . , k + 2 together, etc. Thus when a number of consecutive stations less than k are failed, the telecommunication system still is able to transmit a signal from station 1 to station n. However, if any k consecutive stations fail, the system must fail. Chiang and Niu (1981) introduced a similar example of consecutive 2-out-of n: F system. Example 1.1.2 (Oil pipe system introduced by Chiang and Niu, 1981). A system for transporting oil by pipes from point A to point B has n pump stations. Pump stations are equally spaced between points A and B. Each pump station can transport the oil a distance of k pump stations. If one pump station is down, the flow of the oil could not be interrupted because the next station could carry the load. However, when at least k consecutive pump stations fail, the oil flow stops and the system falls.

240

N. A. Mokhlis

Example 1.1.3 (Railway station introduced by Kuo et al., 1990). A railway station has n lines that receive and send trains. Consider an over size train that requires k consecutive lines in order to enter the station without delay. Then the reliability of the system is the probability that the train enters the station without delay is itself the probability that at least k-consecutive lines which are not in use are available. This is an example of linear consecutive k-out-of-n: G system. Example 1.1.4 (Relayed systems described by Hwang, 1988). In a telecommunication system some object, be it a message, a flow or a signal, is to be relayed from a source to a sink through a sequence of intermediate stations. Case should be taken as to whether the source or the source and the sink are also considered components of the systems, i.e. whether they serve the same functions as the intermediate stations. In telecommunication system source or source and sink and the intermediate stations are all of the same kind of relayed stations. Thus either source or both source and sink are considered components of the systems. In the mobile communication system it is assumed that the source and intermediate stations are all photo-transmitting space-craft bnt not the sink, which could be just an antenna. In these examples where the source (source and sink) is a component of the system, the system fails if the source (source or sink) fails regardless of the value of k. Such a system is an example of relayed consecutive k-out-of-n: F system, unipolar if only the source is included, and bipolar if both source and sink are parts of the system.

1.2. The two-dimensionaI case Salvia and Lasher (1990) introduced the notion of the two-dimensional consecutive k-out-of-n: F systems. Boehme et al. (1992) presented a generalization of such systems as follows. (a) A linear connected X-out-of-(n,m): F(G) lattice system consists of nm components arranged like the elements of (n, m)-matrix, i.e. each of the n rows includes m components and each of the m columns includes n components. The system fails (operates) whenever at least one subset X of connected failed (operating) components occurs which includes failed (operating) components connected in the meaning of connected 2-. (b) A circular connected X-out-of-(n,m): F(G) lattice system consists of n circles, centered at the same point with m rays. The intersections of the circles and the rays represent the elements, i.e. each of the circles includes m components and each of the rays has n components. The system fails (operates) whenever at least orte subset X of connected failed (operating) components occurs which includes failed (operating) components connected in the meaning of connected X. Linear and circular-connected-J(-out-of-(n,m): F(G) lattice systems are generalization of the one-dimensional linear and circular consecutive k-out-of-n: F(G) systems. This is clear by taking X = (1, k), n = 1 and m = n. The subset X could take one of the following forms:


241

(i) (r, s) matrix, r _< n, s < m. (ii) (r, s) matrix or (s, r) matrix, r, s _< n, m, r ¢i s. Connected (r, s)-out-of- (n, m): F(G) and connected (r, s)-or-(s, r)-out-of-(n, m): F(G) lattice systems are found in our daylife in electronic devices, diagnosing diseases, measuring temperature, modeling supervision systems, etc. Example 1.2.1 (Diagnosing disease as presented by Salvia and Lasher, 1990). The presence of a disease is diagnosed by reading an X-ray. The radiologist might not detect the presence of the diseased cells unless they are aggregated into a sufficiently large pattern (say an r x s rectangle). The probability that a small portion (say n x m) of the X-ray is healthy is the same as the reliability of a linear connected (r, s)-out-of-(n, m): F lattice system. Example 1.2.2 (A supervision system was presented by Boehme et al., 1992). A supervision system consisting of 16 TV cameras arranged in four rows and four columns, such that each camera can supervise a disk of radius r. The cameras in each row and column are of the same type and are at a distance r from each other. The supervision system fails if at least two connected cameras in a row or a column fail. This is a linear connected (1, 2) or (2, 1) out of (4, 4): F lattice system. Example 1.2.3 (Measuring temperature system was introduced by Boehme et al., 1992). A cylindrical object covered by a system of feelers for measuring the temperature with five circles each includes four feelers. The measure system fails whenever at least one connected (3, 2) or (2, 3) matrix of failed components occurs. This is a circular connected (3, 2) or (2, 3) out of (5, 4): F lattice system. We shall give a brief survey concerning the exact and approximate formulas as well as bounds for the reliability of one- and two-dimensional consecutive k-outof-n systems. Chao et al. (1995) gave a chronological survey of computing the reliability of consecutive k-out-of-n: F systems.

2. Exact reliability formulas 2.1. Linear consecuti~e k-out-of-n: F s y s t e m s

Assume that the system consists of n linearly arranged independent components. The system falls iff at least k consecutive components fail. Let Pi be the reliability o f / t h component qi - 1 - p » Let Xi denote the state of component i, where X/ = 1 if the component is operating, Xi = 0 if the component is failed. Let X be an n-vector such that the component i of X is 1 or 0 depending on whether component i of the system is operating or not, respectively. Chiang and Niu (1981) presented a recursive formula to compute the exact system reliability by defining the vector X of component states and a random

N . A . MokhlB

242

variable indicating the index of first 0 in the vector of states. They gave recursive equations requiring O(n2k) computing time. The derivation of the recursive formula is based on examining the first sequence of consecutive 0's in the X vector. If the number of consecutive 0's in the first sequence is at least k, then the system is failed. If the number of consecutive 0's in the first sequence is less than k, then the reliability of the system is equal to the reliability of a consecutive k-out-of-n': F system where n' < n. Let Z be a random variable indicating the index of first 0 in X, and M be a random variable indicating the index of first 1 after position Z in X. The reliability of the system is RLF(k; n;p) = Pr {the system is operating} = Z

Pr{the system is operatingIZ = z,M = m}

Z

z

m

Pr{Z = z , M = m} . When Z > n - k + 1, the system will have less than k failed components and Pr {the system is operatingIZ > n - k + 1} = 1 and Pr(Z > n - k + I) = ph-k+1. When m > z + k, the system will have already consecutive k failed components, and the system is failed. Thus n k + l z+k-1

RLv(k;n;p) = ~ z--1

Z

Pr{the system is operatinglZ = z,M = m}

m=z+l

x Pr{Z = z , M = m} +ph k+~ For z + 1 < m < z + k - 1, the first sequence of failed components does not constitute a cut set. Furthermore, since Xm = 1, the event that consecutive k-outof-n: F system is equivalent to the event that a consecutive k-out-of-(n - m): F system is operating. It is clear that Pr{Z = z, M = m} = prqm-r where q = 1 - p. Thus, the recursive formula for RLF(k; n;p) is n - k + l z+k 1

RLv(k;n;p)= ~ z=l

RLF(k;j;p) =

{1,

0,

Z

RLF(k;n--m;p)Prqm-r +p" ~+~ '

(2.1)

m--z+l

O<j_2 ,

(2.3)

i=O

N(j,r;1)=

I

(«), O«_jr

.

For the special case k = 2, [n+1/2] ~ (

RL(2, n;p) = ~-~

)

n - j + 1 pn-jqj J

(2.4)

.

Eq. (2.4) was derived by Chiang and Niu (1981) by considering the number of ways that, o f j failures and ( n - j) successes, at least one success can be placed between every two failures. Hwang (1982), derived two sets of recursive equations for the reliability of the system even when components are not necessarily identical, by two different arguments. Let Ei denote the event that component i is the last operating one, and F~ denote the event that the system first falls at component i. Clearly E i ' s a r e disjoint events and exactly one of the events En-k+1,.. ,E,, taust occur for the system to operator. Thus we have

RLF(k;n;p~, ..,pn) = ~

P(Ei)RLF(k;i- 1;pl,... ,Pi-1)

i~n-k+l

=

Pi

RLF(k;i- 1;pl,.

i=n-k+ l

I qj

,Pi-1) ""

(2.5) with the boundary condition

RLv(k;n;pl,...,pn) = 1 for n < k . For identical components Eq. (2.5) becomes RLF(k; n;p)

= ~ i=n-k+l

pqn - i R

LF(k;

t" - -

1;p)

.

(2.6)

244

N. A. Mokhlis

A better a p p r o a c h is to w o r k with F~. In the definition of F~, i is the smallest subscript such that the k consecutive c o m p o n e n t s i - k + 1, i - k + 2 , . . , i - 1, i fail. In particular, if i > k, then F / i m p l i e s that c o m p o n e n t (i - k) is operating. Since Fi's are disjoint events and one of them must occur for the system to fail, we have //

1 -- RLF (k; n ; P l , - . . ,

Ph) = Z P(Fi)RLF (k; i -

k - 1; P l , . . . ,

Pi-k-1 )

i=k n

i

= ~-~RLv(k;i-k-1;pl,...,Pi-k-~)pi-k i=k

II

«J

(2.7)

j=i-k+l

with b o u n d a r y conditions

po-l,

RLF(k;n;pl,...,pù) = I

forne(k;

n;pl,...

,p,~) = RE(k; n - 1 ; p l , . . . ,P,,-I)

- RLF(k;n - k - 1;pl,...,pn-k-1)pn-k

~I

qj .

(2.8)

j=n-k+l

F o r identical c o m p o n e n t s

1 --RLF(k;n;p) = I1 - R L ( k ; n -

1;p)]

+RL(k;n-k-

1;p)pq k .

(2.9)

Lambiris and Papastavridis (1985) derived an expression for the numbers NO'; r; k - 1), r _> 0 in Eq. (2.2), by using their generating function

9(t)=\l-tJ

(a=~o(2)

(i=~o (2.10)

=~ß(:)(r+J-k2-1)(-l)atJ j=k;~a=0 j -- k2

245

Consecutive k-out-of-n systems If we set in (2.10), r = n - j ÷ 1, the coefficient of

tJ is

~'+'( )( j_k),B(-1) ~ n - j +~ l n'«

N(j;n-j+l,k-1)=

;~ .

2=0

(2.11)

Substitution of (2.11) into (2.2) yields

ù~~l) in which akj = pnkjqk-nk«, (n2,1, n2,2, n2,3, n2,4) = (2, 1, 1,0), ~ nk-lj+l nkj = I. nk 1,j

for k _> 3 and for k _> 3 and

1 < j _< 2 k-l, 2 k-1 q- 1 _< j _< 2 k ,

Qk is (2 k - 1) x (2 k - 1) matrix obtained by deleting both the last column of the 2 k x 2 k matrix

pk(i,j)=

{

P q

0 pk(i -- 2k-l,j)

for for for for

1 < i < 2 k-l, j=2i1, 1 < i < 2 k-l, j = 2i, 1 < i < 2k-l, j e 2 i - 1,2i, 2 k 1 + 1 < i < 2 k, 1 _<j_ n,

1) -~ q(2, 1), (2, 3)) = pRLF(2; 5;p) + qp3 = p6 + 5pSq + 6p4q2 +p3q3 + qp3 . The reliability of a circular-connected-(l, 2)-or-(2, 1)-out-of-(2, 3): F lattice system is obtained by searching all the failure stares of the system. Thus, Rcv((1,2)-or-(2, 1), (2, 3)) = 1 - (9p4q 2 q- 20p3q 3 + 15p2q4 + 6pq 5 + q6) . The reliability of a linear connected-(1, 2)-or-(2, 1)-out-of-(3, 3): F lattice system is obtained by relating it to the reliability of a one-dimensional circularconsecutive 2-out-of-8: F system. That is RLv((1,2)-or-(2, 1), (3, 3)) = pRcv(2, 8;p) +p3q . Mokhlis et al. (1998) applied a different approach for obtaining exact formulas for the reliability of the more general systems, linear (circular)-connected-(1, 2)or-(2, 1)-out-of-(n, 3): F lattice systems. Their approach depends on counting the operating states of the system. The reliability of a linear-connected-(1, 2)-o>(2, 1)-out-of-(n, 3): F system is given by the following recursive formula: RLF((1,2)-or-(2, 1), (n, 3)) n+l [j/2]

= ~-~~Zfiij(+i-ip 2j (i-1)RLv((1,2)-or-(2 , 1), (n --j, 3)) ,

(2.47)

j - 1 i=0

where RLz((1,2)-or-(2, 1), (-1, 3)) = p 3 B01= 1, f i 0 j = 3 ' U -2 for l < j _ < n + l

(2.48) ,

j-I

Bij=

~

C~mqm(j) for 1 < i
(2, 1), (--1,3)) = q-3 and fl« as in (2.48) (2.50). (h) Reliability of circular-connected-(1,2)-or-(2,1)-out-of-(n,3): G lattice systems: RCG((1,2)-or-(2, 1), (n, 3)) = 1 --PcG((1,2)-or-(2, 1), (n --j, 3)) , where pCG((1,2)-or-(2, 1), (n, 3)) n+l

= Z YJpj-lq2j+lpcG((I' 2)-or-(2, 1), (n -- j, 3)) j=l

7 j = 3 " 2 j-2

for j_>2, 71 = 1

and PCG((1,2)-or-(2, 1), (--1, 3)) = q-3

2.6. Algorithm for computing the reliability Bollinger (1982, 1986) introduced a direct computation and an algorithm for calculating a table of coefficients for the failure probability polynomials associated with a linear consecutive k-out-of-n: F system with i.i.d components, depending on the properties of the list of the 2n binary strings. Following the same idea E1Sayed (1998) introduced an algorithm for calculating the coefficients for the failure probability of a (1, 2)-or-(2, 1)-out-of-(n, 2): F lattice system.

N. A. Mokhlis

264

Khamis and Mokhlis (1997) derived an algorithm for computing the reliability of linear and circular connected- (1, 2)-or-(2, 1)-out-of-(m, n): F lattice systems. In 1998 they derived an algorithm for enumerating the reliability of linear and circular connected-(r, s)-out-of-(n, m): F lattice systems as well as the reliability of linear and circular connected-(r, s)-or-(s, r)-out-of-(n, m): F lattice systems with i.i.d components.

3. Approximation formulas and bounds

3.1. One-dimensional consecutive k-out-of-n: F systems Chao and Lin (1984) examined the limiting behavior of RLF(k; n;p). From (2.17) it is clear that for fixedp, RLF(k; n;p) --+ 0 as n ~ oc. Also, irk _> 2 a n d p ~ 1, for example, with n(1 - p ) -+ 2, RE (k; n;p) ---, 1. Neither case is interesting for system design purposes. They showed that the rate of convergence depends on k. They proved that for 1 < k < 4 lim RLF(k;n;p) = exp(--nqk),

n---+OQ

provided that nq k is bounded .

(3.1)

Recall the stationary Markov chain Y1, Y 2 , . . , where Y/=(Xi,~/+l,...,X/+k_l) ,

i = l,2, . . , n - k + l ,

where X/is the state of component i, X~ = 1 or 0 if the component is functioning or failed, respectively. If S, is the number of times that Y,- enters the taboo set, then Sù converges to Poisson with rate nq k. Chao and Lin (1984) conjectured that (3.1) holds for k > 4. Fu (1985) proved that (3.1) holds for k > 1. The technique for proving this result is based on evaluating upper and lower bounds of RLF(k; n;p). For every small 6 > 0, there exists no(Ô) such that for all n > no(6)

(

2k 1

-

-

,

(1+6)2k+, ~-~o(,)

~

U+7i77~

"~

-)

>- eLf(k; n;p) >-

where

q = 2n-~/k,

{

k

no(f) =

n*(6) n

ifn*(6) < k, ifk_< n*(6) < n, ifn*(6) > n,

(

1-

~ ) ~ k+1 '

(3.a)

265


Fu and Hu (1987) studied the reliability of linear consecutive k-out-of-n: F system when the component failure states have ( k - 1 ) step Markov dependence. This means that the failure probability qi and reliability pi of component i (i = 1 , 2 , . . , n ) depends on the number of m consecutive failures immediately preceding this component, i.e., qi = q(m), Pi = p(m) and

q(m) + p(m) = 1 Recalling (2.22) and (2.23) we have RLF(k;n;p(m)) = S

q(j)

m=0\j=0

~-~~p(r)R(r,k,n - m - 1) ,

r=0

where R(m; k, i) is the probability that the system is working at component i with ending m consecutive failures, and is given by

R(m; k; i)

(m~)~~ q(j)

p(r)R(r; k; i - m - 1 ) ,

1 < m < k -1

r=O

and k-1

R(0;k;i) = Z p ( r ) R ( r

; k ; i - 1) .

r--0

The following theorem stares that if the failure probabilities m = 0, 1 , . . , k - 1 are functions o f n and tend to zero with rates

q(m) = 2mn -I/k,

2m>0

q(m),

form=0,1,..,k-1

then the reliability of the system tends to a given constant.

THEOREM. If q(m)=2mn

z/k, m = 0 , 1 , . . , k - 1

,

(3.3)

then RLF(k; n; p(m) ) ~ exp I --n k-1 I~_oq(m )

(3.4)

and lim

n---+o~

RLF(k;n;pl,...,pn) = exp -

2m

•

(3.5)

For )~m = 2, m = 0, 1 , . . , k - 1, the dependence disappears and (3.5) is reduced to (3.1). Many researchers have proposed lower and upper bounds for the reliability of linear and circular consecutive-k-out-of-n-systems. For example, Chiang and Niu

N. A. Mokhlis

266

obtained lower and upper bounds for linear consecutive-k-out-of-n: F system with identical components operating s-independently given by

(3.6)

(1 - qk)n-k+l < RLv(k; n;p) < (1 - qk)[~/k]

Derman et al. (1982) generalized (3.6) to the linear and circular consecutive kout-of-n: F systems with independent but not necessarily identically distributed components failure probabilities. They showed that R~F(k; n;pl,... ,p~) for 7 = L or C satisfy

Il 1- H i= 1

«~

j=i

/

-< R,~(~;ù;p~,..,pù),

(3.7)

where

g= { n- k +

for7=cf°rT=L'

and

Æ2(N)

(3.8)

RyF(k; n;pl,... ,p~) 0.95, f o r p < 0.95 the best upper b o u n d s are i and U~c for linear and circular systems, respectively. U3L F r o m what is mentioned here above, we conclude that best lower and upper bounds depend on the values o f p, r, s, n and m. It is clear that the reliability o f an F system is greater than that o f a circular F system. Also, we see that the reliability o f (r, s)-out-of-(n, m): F system is greater than that o f (r, s)-or-(s, r)-out-of-(n, m): F system. This is because, the n u m b e r o f minimal cuts of circular systems is greater than that o f linear systems, as well as the n u m b e r o f minimal cuts o f (r, s)-or-(s, r)-out-of-(n, m) is greater than that of (r, s)-out-of-(n, m): F systems. 3.3. Consecutive k-out-of-n: G systems

Using the relation that consecutive k-out-of-n: G and consecutive k-out-of-n: F systems are mirror images o f each other K u o et al. (1990) derived bounds for the one-dimensional linear and circular consecutive k-out-of-n: G systems.

275

C o n s e c u t i v e k-out-oj~n s y s t e m s

v

~L ©

H

ù

? Y

3

~5 ©

z

Il

©

& ©

N . A . Mokhlis

276

Il r~

.=

0

9

? Y

ù.-,

~eq

3ù

~5

II

-8 ùt ©

t-'q

8 ¢.~

8 M

~


tt~

©

H ,A

ù~

©

? ,..y Y ee~

Il

ù

©

277

278

N. A. Mokhlis

o@ ¢'~

o

~~

oo

ddd

©

t~

gù.ù

t~

r~


279

The lower bounds are g~=l-

H

1-

PJ

H

for•=LorC.

j=(i-l)k+l

i=1

The upper bounds are g (

~,= l - II

i=1

i+k- 1 )

1 - II pJ j=i

for7

LorC

,

/

where

g={n-k+l n

for 7 = L, for 7 = C .

The bounds for the reliability of the two-dimensional: F systems could by used by means of the mirror image relation, to obtain bounds for the corresponding G systems. References Antonopoulou, I. and S. Papastavridis (1987). Fast recursive algorithm to evaluate the reliability of a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-36, 83-84. Arratia, R., L. Goldstein and L. Gordon (1989). Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Prob. 17, 9-25. Barbour, A. D., L. Holst and S. Janson (1991). Poisson Approximation. Oxford University Press. Barlow, R. E. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing. Holt, New York. Boehme, T. K., A. Kossow and W. Press (1992). A generalization of consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 41, 451~457. Bollinger, R. C. (1982). Direct computation for consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-31, 44~446. Bollinger, R. C. (1986). An algorithm for direct computation in consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-35, 611 612. Bollinger, R. C. and A. A. Salvia (1982). Consecutive-k-out-of-n: F networks. IEEE Trans. Reliab. R-31, 53 56. Chao, M. T. and J. C. Fu (1989). A limit theorem of certain repairable systems. Ann. Ins. Stat. Math. 41, 809-818. Chao, M. T. and J. C. Fu (1991). The reliability of large series systems under Markov structure. Adv. Appl. Prob. 23, 894-908. Chao, M. T., J. C. Fu and M. V. Koutras (1995). Survey of reliability studies of consecutive-k-out-ofn: F and related systems. IEEE Trans. Reliab. 44, 120-127. Chiang, D. T. and R. Chiang (t986). Relayed communication via consecutive-k-out-of -n: F system. IEEE Trans. Reliab. R-35, 65 67. Chiang, D. T. and S. Niu (1981). Reliability of consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-30, 87 89. Chao, M. T. and G. D. Lin (1984). Economical design of large consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. R-33, 411-413. Chrysaphinou, O. and S. Papastavridis (1990). Limit distribution for a consecutive-k-out-of-n: F systems. Adv. Appl. Probab. 22, 491493.

280

N. A. Mokhlis

Derman, C., G. J. Liberman and S. Ross (1982). On the consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-31, 57-63. E1Sayed, E. M. (1998). Algorithm for reliability of-(1, 2)-or-(2, 1)-out-of-(n, 2): F system. J. Egypt. Math. Soc. 6, 169-173. Feller, W. (1968). An Introduetion to Probability Theory and Its Applications (3rd ed.). Vol. 1, Wiley. Fu, J. C. (1985). Reliability of a large consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-34, 127-130. Fu, J. C. (1986a). Bounds for reliability of large consecutive-k-out-of-n: F system with unequal component probabilities. IEEE Trans. Reliab. R-35, 316 319. Fu, J. C. (1986b). Reliability of consecutive-k-out-of-n: F system with (k-1) step Markov dependence. IEEE Trans. Reliab. R-35, 602-606. Fu, J. C. and B. Hu (1987). On reliability of a large consecutive-k-out-of-n: F system with (k - 1)-step Markov dependence. IEEE Trans. Reliab. 36, 75-77. Hwang, F. K. (1982). Fast solutions for consecutive-k-out-of-n: F related systems. IEEE Trans. Reliab. R-31, 447448. Hwang, F. K. (1988). Relayed consecutive-k-out-of-n: F lines. IEEE Trans. Reliab. 37, 512-514. Hwang, F. K. (1993). An O(k n)-time algorithm for computing the reliability of a circular consecutivek-out-of-n: F system. IEEE Trans. Reliab. 42, 161-162. Khamis, S. M. and N. A. Mokhlis (1997) An algorithm for computing the reliability of connected(1, 2)-or-(2, 1)-out-of-(m, n): F lattice system. Congressus Numerantuim 127, 143-154. Kontoleon, J. M. (1980). Reliability determination of a r-successive-out-of-n: F system. IEEE Trans. Reliab. R-29, 437. Koutras, M. V., G. K. Papadopoulos and S. G. Papastavridis (1993). Reliability of 2-dimensional consecutive-k-out-of-n: F systems. IEEE Trans. Reliab. 42, 658-661. Kuo, W., W. Zhang and M. Zuo (1990). A consecutive-k-out-of-n: G system: The mirror image of a consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 39, 244253. Lambiris, M. and S. Papastavridis (1985). Exact reliability formulas for linear and circular consecutive-k-out-of-n: F related systems. IEEE Trans. Reliab. R-34, 124-126. Malinowski, J. and W. Preuss (1996). Lower and upper bounds for the reliability of connected-(r, s)out-of-(m, n): F lattice systems. IEEE Trans. Reliab. 45, 156-160. Mokhlis, N. A. and A. S. Mohamed (1999). Bounds for reliability of relayed consecutive-k-out-of-n: F system. J. Egypt. Math. Soc. 7(2), 225 237. Mokhlis, N. A. and S. M. Khamis (1998). An exact enumeration of the reliability of some linear and circular connected-X-out-of-(n, m): F lattice system. Congressus Numerantium 135, 93 118. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1997). Bounds on reliability of connected-(r, s)-ont-of(n, m): F lattice systems. In The 32-nd Annual Conference on Statistics and Computer Science Operation Research. Vol. 32, pp 19-34. I.S.S.R., Cairo University. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1998). Reliability of a connected-(1, 2)-or-(2, 1)-outof-(n, 3): F lattice system. J. Egypt. Math. Soc. 6, 175 183. Mokhlis, N. A., E. M. E1Sayed and G. Youssef (1999). Reliability of consecutive-k-out-of-n systems. M.Sc. Thesis. Papastavridis, S. (1987). Upper and lower bounds for the reliability of consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-35, 607 610. Papastavridis, S. and M. Lambiris (1987). Reliability of a consecutive-k-out-of-n: F system, for Markov-dependent components. IEEE Trans. Reliab. R-36, 78-79. Salvia, A. A. and W. C. Lasher (1990). 2-dimensional consecutive-k-out-of-n: F models. IEEE Trans. Reliab. 39, 382 385. Wu, J. S. and R. J. Chen (1992). An O(k n) algorithm for a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. R-41, 303 305. Wu, J. S. and R. J. Chen (1993). Efficient algorithm for reliability of a circular consecutive-k-out-of-n: F system. IEEE Trans. Reliab. 42, 163-164. Zuo, M. J. (1993). Reliability and design of two-dimensional consecutive-k-out-of-n system. IEEE Trans. Reliab. 42, 488-490.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserve&

| •

lk &

Exact Reliability and Lifetime of Consecutive Systems

Sigeo Aki

1. Introducfion

A linear consecutive-k-out-of-n: F system is a system of n components in sequence where the system fails if and only if k consecutive components fail. The system was introduced by Kontoleon (1980). The reliability of the system has been investigated by many researchers (e.g., Chiang and Niu, 1981; Derman et al., 1982; Shanthikumar, 1982; Lambiris and Papastavridis, 1985; Hwang, 1986; Papastavridis and Hadzichritos, 1988; Peköz and Ross, 1995). Excellent surveys were presented by Hirano (1994) and Chao et al. (1995). For i = 1 , . . , n, let X/be a random variable such that X / = 1 (= 0) if the ith component is functioning (resp., fails). When X1,X2,..,X~ are independent identically distributed (i.i.d.) random variables with P ( ~ = l ) = p and P(Xi = 0) = q(= 1 - p ) , the reliability of the system is given by

rk)--(n-k(r+l))(-1)qk)

R(p) = Z ( - 1 ) r ( p q k ) r r=0

.

F

Many papers have been published in reliability journals and in most papers the reliability value of each consecutive system was calculated under various conditions mainly by means of techniques of enumerative combinatorics. In view of practical situations, some generalizations of the system have been attempted. For example, some dependence structures between components have been taken into consideration (Fu, 1986; Papastavridis and Lambiris, 1987; Ge and Wang, 1990) and more complex systems such as m-consecutive-k-out-of-n: F system (Papastavridis, 1990), consecutive-k-out-of-r-from-n: F system (Griffith, 1986; Sfakianakis et al., 1992) and some extensions to two-dimensional systems (Salvia and Lasher, 1990; Ksir, 1992; Boehme et al., 1992; Yamamoto and Miyakawa, 1995) have been investigated. As the systems become more complex, it seems more difficult to calculate their exact reliability values by means of usual techniques of enumerative combinatorics. However, we can deal with the problem successfully by using new powerful tools such as Markov chain imbedding method and the method of conditional 281

282

S. Aki

probability generating functions, which have been recently developed in statistical distribution theory of runs. Statistical distribution theory of runs has also developed extensively over the past two decades. In particular, the exact distribution of the number of runs of a specified length in various random sequences and that of the number of trials until the first occurrence of a run of a specified length in a sequence of trials have been investigated and derived by using the above new tools even if the sequence of trials is dependent or multivariate. Actually, the reliability value of each consecutive system means the probability that a run of failed components of a specified length does not exist in the sequence of components. Therefore, we can use some results in distribution theory in order to calculate the reliability values of much more complex consecutive systems. Here, we briefly overview some approaches to the study of discrete distributions of runs, which are substantially related to the reliability study of consecutive systems. Of course, the standard method is enumerative combinatorics also in the study of statistical discrete distribution theory. The method is the following: to start with, all the typical sequences are considered, and then, they are split into subsequences which can be interpreted successfully. The method is useful when trials are independent or nearly independent (e.g., when the sequence of trials is a Markov chain). However, when dependence structure of trials becomes complex (e.g., when the sequence of trials is a higher-order Markov chain), it becomes difficult even to write down all the typical sequences (see Aki et al., 1996). The Markov chain imbedding method is to imbed the sequence of trials (or enumeration process of runs or patterns) into a Markov chain with an appropriate state space. Then the desired probability of the number of runs or patterns can be obtained by multiplying the transition probability matrices. The method was introduced by Fu and Koutras (1994) and developed by Fu (1996), Koutras and Alexandrou (1995, 1997a, b) and Koutras (1997). The method of conditional probability generating functions (p.g.f.) can be explained as follows: to begin with, all the different situations which are possible to occur are considered, and the relations between conditional p.g.f.'s given the situations are examined, and finally, the system of equations which represent the relations is solved. The method was introduced in statistical distribution theory of runs by Ebneshahrashoob and Sobel (1990) and used in various situations by Hirano and Aki (1993), Aki and Hirano (1995), Uchida and Aki (1995), Balakrishnan et al. (1997), Hirano et al. (1997) and Han and Aki (1998). In Section 2, we explain how to use the method of conditional p.g.f.'s. When the relations between conditional p.g.f.'s are linear, the method and the Markov chain imbedding method are essentially equivalent. Markov chain imbedding approaches for reliability study of usual consecutive systems were investigated very well by Koutras (1996). Hence, in order to emphasize a merit of the method of conditional p.g.f.'s we deal with an example of a consecutive system on a directed tree, where relations between the conditional p.g.f.'s are not necessarily linear. In Section 3, we give some numerical examples of the reliability of consecutive systems on directed trees. Algorithms for computing the p.g.f.'s of the number of

Exact reliability and lifetime of conseeutive systems

283

failure runs of a specified length k are also given. By using the algorithms and computer algebra systems, we can obtain the exact p.g.f.'s. After expanding the p.g.f, with respect to the variable t (if necessary), we can easily calculate the exact reliability of the k-consecutive system by substituting t = 0 into the p.g.f. In order to calculate the reliability of the m-consecutive-k-out-of-n: F system on a directed tree, we only sum up the coefficients of tJ of the p.g.f, for j = 0, 1 , . . . , m - 1. In Section 4, we discuss lifetime distributions of consecutive systems. After giving a very general result, we review lifetime distributions of some systems systematically. In particular, we focus on derivation of the mixing weights by which the lifetime distribution of a consecutive system is represented as a mixture of distributions of order statistics of lifetimes of the components.

2. Reliability of consecutive systems on directed trees Let us consider a consecutive system on the directed tree in Figure 1. The root of the tree is vl and all the edges are directed away from the root. Each vertex is assumed to be a component of a system, which fails if and only if three consecutive components along the direction fail. For simplicity, we assume also that all the components independently fail with probability q (= 1 - p ) . By using the method of conditional p.g.f.'s, we can derive the exact distribution of the number of non-overlapping failure runs of length three along the direction on the directed tree and calculate the reliability value of the system. In the i.i.d, case, the following result holds generally. We fix any vertex v. Let T~ be the subtree which consists of the vertex v as the root of Tu and of all the descendants of v. In the path of all the ancestors of v from the root to the parent of v, assume that a failure tun of length g along the direction is observed at the parent of v. If the vertex v is the root, we conventionally define that a failure run of length 0 is observed at the parent of v, though the root does not have its parent. 731

U4

U5

~, U6

\

/ /\ 739

7310

7311

U12

V13

U8

'/)14

Fig. 1. An example of a directed tree.

s. Aki

284

Then we denote by 4(/),g;t) the p.g.f, of the conditional distribution of the number of non-overlapping failure runs of length k along the direction in the subtree T~. If the vertex v has c(v) children/)(1), v ( 2 ) , . . , v(c(/))), the numbers of failure runs of length k in T~(1), T~(2),. • •, and T~(c(~)) are conditionally independent given that all the ancestors from the root to the vertex/) are observed. Hence, by considering one-step ahead from the vertex v, we obtain the following proposition. PROPOSITION 1. If all the components independently fail with probability q(= 1 - p ) , the conditional p.g.f.'s satisfy the recurrence relations:

4(v,g;t) = ~ lPY[~(--~Iq~(v(j),o;t)+ qtI]:(=vlqS(v(j'),0;t) /

(p+qt

ifc(v) > 0 and g < k ifc(v) > 0 and g = k ifc(v) =0 and g < k ifc(v)=0 and g = k -

1, 1, 1, 1.

(1) REMARK. In this section we assume that all the components fail independently with a common probability. However, more general results can be derived. Aki (1999) treated the problem when the joint distribution of the components is a directed Markov distribution (see, e.g., Lauritzen, 1996). He obtained the exact distribution of the number of non-overlapping runs on directed Markov trees. In fact, Proposition 1 is a special case of the result of Theorem 2.1 of Aki (1999). However, it should be noted that the result of Proposition 1 is written very simple because of the assumption. By using Proposition 1, we shall derive the p.g.f, of the number of non-overlapping failure runs of length three along the direction on the directed tree given in Figure 1. For simplicity, it is assumed that all the components fail independently with probability q (= 1 - p ) . Let 4(t) be the p.g.f. Then, 4(t) = 4(Vl, 0; t) holds from the definition. By applying (1) to the root/)1, we obtäin

4(/)~, o; t) = p4(/)2, o; t)~(V3, O;/)4(V4~ O; t) + q4(/)2, 1; t)~(~3, l; t)~(~4,1; t) . Next, by applying (1) to every child of the root Vl, we see that

4(~2, o; t) = p4(/)», o; t) + q4(/)», 1; t), 4(/)2, 1; t) ~-~P445, 0; t) + q4(/)5, 2; t), 4(/)3, 0; t) = P446,0; t) -k q446, 1; t), 4(/)3, 1; t) = P4(Vo, 0; t) q- q4(v6, 2; t) , and

~(V4,0; t) ~ p 4 ( v % 0; t)4(V8,0; t) + q4(v7, 1;t)4(V8, 1; t), 4(v4, 1; t) = pO(v7, O; t)4(v8, O; t) + q4(v7, 2; t)4(vs, 2; t) .

285

Exact reliabiIity and lifetime of consecutive systems

Since the vertices v9,Vl0, Vll,Vl2, Vl3 and /)14 do not have a child, we see from Proposition 1 for i = 9, 1 0 , . . , 14, Ó(vi, O;t)= l,

O(vi, 1 ; t ) = l,

é(vi,2;t)=p+qt

.

Then, by applying Proposition 1 to the vertices v», v6, v7 and va, we obtain

qS(vs, 1; t) = p + q(p ÷ qt) 2, ~b(v6, 0; t) = 1, qS(v6, 1; t) = p q- q(p + qt) 2, ~b(v7, 0; t) = 1, qS(v7, 1;t) = p + q ( p + q t ) , é(vs, 0; t) = 1, ~b(v8, 1; t) = p + q(p + qt),

qS(vs, 0; t) = 1,

qb(vs, 2; t) = p + qt, qS(v6, 2; t) = p + qt, qS(v7, 2; t) = p + qt, OS(va,2; t) = p + qt .

Therefore, we obtain ~b(t) = p~o + q{p + q(p ÷ qt)2}12~ + q{p + q(p + qt)} 2] + q{p + q(p + «t)}2{p + q(p + qt) 2} . The reliability value of the consecutive system is the coefficient of to in the polynomial qS(t). Hence, we can write the reliability value of the consecutive system as p(p + q(p + qp2))2(p + q(p + qp)2) + q(p + qp)2(p + qp2) . As a corollary of Proposition 1 we can deal with a linear consecutive-k-out-of-n: F system. We can regard the special tree given in Figure 2 as the system. According to the result of Proposition 1, we see that the p.g.f.'s of the conditional distributions of the number of non-overlapping failure runs of length k satisfy the linear recurrence relations: [p4)(vi+l,0;t)+qcB(vi+l,g+l;t) O(vi,g;t)= ~pß(vi+l,0;t)+qt~(Vi+l,O;t)

l"

kp+qt

ifi 1 and 6 < k - 1, i f u > l and g=k-1, ifu=l and6_v,

1 _v,

1 1"

--

1; hence

Var~*

~

cv + 2

(cv --

2 G ~2 c~ > 2; 1)2(Cv -- 2) v,

2

M S E ( ~ ~ ) = («~ - - i ) 7 7 , - 2) ~~'

«v > 2,

1 < v < r~ .

Sequential k-out-of-n systems

319

4. The s«atisti« (Œ*l,..., c~*~,) is «ompl«tely suffi«i«nt f o r ( o q , . . . , c~~,). 5. The sequences o f estimators (c~~) are strongly consistent

w.r.t.

Cv --+ O(3(1 < V < F1).

6. c~; is asymptotically normal, 1 < v < rl, i.e., v@7(c~~/~v - 1) ~ Cv ---+(X).

JV'(0, 1) w.r.t.

In part 2 of the the above theorem it is shown that Œ~ is distributed as lITt, where the random variable T~ is gamma distributed with parameters cu and (c~ • c~~)-1, i.e, its density function is given by fr~(t ) -- («vc~~)~~ t«,,_le_«~~#

(7~ --F)!

t _> o

The distribution of c~; is known as inverted gamma distribution, which is used as a prior density in Bayesian analysis (cf. Bain, 1983; Johnson et al., 1994, p. 524).

4.2. Identically structured systems

Let us consider s _> 1 iid observations of some sequential (n - r + 1)-out-of-n system leading to the set of data (xu)~ 0, c --- 1/d), and in line 5 we have log-logistic distributions (a > 0, c = -I/d). Based on F E ~-, the joint distribution of X,(1),..,X, (4 is given by the Lebesgue density function fx!il"~r(*~l (xl , . . . ,x~)

z

(,

-Il! ,')!" ~

=

I O~i riI~ll[O;ig (xi) exp{-~(n-i

+ 1)(g(xi)-g(Xi_l)

) }] (6.1)

where g-1 (#) = xo ~ Xl ~ ... ~ Xr" For g(t) = t, this distribution coincides with the distribution of the first r (out of n) order statistics based on a Weinman multivariate exponential distribution (cf. Weinman, 1966; Johnson and Kotz, 1972, p. 268/9; Block, 1975, p. 303), which in Cramer and Kamps (1997b) and Cramer (2001) is referred to as WMEn(#, 0, ~), = ( c q , . . . , c~,). This particular multivariate exponential distribution is an extension of Freund's bivariate exponential distribution (cf. Freund, 1961). Hence, our results contribute to the analysis of Weinman's multivariate exponential distribution. Regarding related works in estimation theory, we are only aware of point estimation results for the parameters c~j given by Weinman (1966) and for the entropy of the distribution (6.1) with g(t) = t (cf. Ahmed and Gokhale, 1989). Putting ~1 . . . . . 0~r = 1 in (6.1), we obtain the joint density function of ordinary order statistics X I # , . . , Xr,n based on iid random variables 321,.. ,X, with distribution Exp (#, zg). For F ~ ~ as an underlying distribution, sequential order statistics can be introduced in a simplified manner as compared with Definition 2.1. Given F ~ ~with parameters # and ~9 we find F-l( t)=g

l(#_Olog(l_t)),

tel0,1)

.

Assuming (1.1), i.e., Fr = 1 - (1 - F) ~r, 1 < r < n, we conclude that Fr c ~- with the same function g and parameters # and 0/c~r, 1 < r < n. This set-up can be interpreted as follows. If r - 1 components have failed, the residual lifetimes of the remaining components are supposed to be distributed according to N c Y with scale parameters 0/~r, 2 < r < n. Let (Y)i/)l0, a > 0 ,

0>0

.

It is denoted by Wei(0, a). Kamps and Cramer (1999) have calculated the moments of order v of sequential order statistics based on two-parameter Wei(O/cq, a)-distributions provided that (n - i + 1)~i • (n - - j + 1)czj for i C j, 1 _ 2 -Iv/a+l) ,

333


where r

aj=aj(r)=II((n-k+l)ak-(n-j+l)c~j)

-1

1 <j0 ,

a recurrence relation for the moments results from formula (2.5) in Cramer and Kamps (2000): 0(v+a)

E (X •fr)) v+a -E(X(f-1)) ~+a =

a(fi_--;TT)~

L,;v(r)~~ ~~l~,

j ,

~ >

-a

.

For v = 0 the formula reduces to E(X(r))a - E ( X ( r - l ) ) a

-- (n - r + 1)at "

A recurrence relation for the product moments of the three-parameter Weibull distribution is shown in Cramer and Kamps (2000, Example 3.5).

6.4. Moments of sequential order statistics based on Pearson distributions

(( ~/1~~ )

In Table 6, line 4, distribution functions of the form F(t)=l-(dO(c-ta))

1/~,

tE

c-

,c 1la

a>0,

,

d>0,

v~>0

are introduced including special Pearson I and power function distributions. For these distributions we find the identity E(X~r))v+a

E(x(,r 1))v+a

v+a

(~--~ ( n - j + l ) ~ j

= a d ( ( n - F~- 1)~ r-~~9) \«=1 ( n - - 7 ~ ~ ) ~ j T ~ 9

) E(y*(r))v '

where y.(1),.., y.(•) are sequential order statistics based on E" = 1 - (1 - F ) ~i, 1 0

Sequential

k-out-of-n

335

systems

7.1. E s t i m a t i o n in the l o c a t i o n - s c a l e f a m i l y

We consider both the case of a known parameter # and that of unknown # and 0. If p is supposed to be known, g may depend on #. By analogy with Theorem 4.1 we obtain THEOREM 7.1 • T h e M L E s o f ~ T h e o r e m 4.1, i.e.,

= T, ~ 1 < v < rl, are g i v e n b y ~~* = T~ with c~v* as in

-1 O~1 =

S

ni[g

) - - [A

\i=!

and -1

-,

)

ni-v+l)[g

1)

)-g

)1

,

2 o.

02 Hence, O* 2. F(~~,~k ~~~ j -- ~(Æ+k-1)!(~)k ~~/ , »,~ E N; in particular, EO* = 0 and Var O* -- -y. is an unbiased estimator of ~9. 3. O* is sufficient for O. 4. (O*(R)) R is strongly consistent for #, i.e., #*(R) -+ 0 a.e. w.r.t. R -+ cx~. 5. (O*(R)) R is asymptotically normal, i.e., x / R ( O * ( R ) / O - 1 ) d j v ( 0 , 1) w.r.t.

R---+ oo.


339

In the situation of type II censoring, i.e., cqj = 1, ri = r, ni = n, 1 0, # = 0). In case of the exponential distribution Basu and Singh (1998) present the M L E of 0 for s _> 1. The distributional results of Theorem 8.2 can be utilized to derive confidence intervals for the parameter ~. From the first item of the above theorem we find ~9" ~ ~5{2R,~°2where Zq2 denotes the z2-distribution with q degrees of freedom. Hence, a two-sided confidence interval with level ~ E (0, 1) is given by

[

. 2RO* 2RO* ] )~2R(1 - ~/2)' )~22R(cz/2)J '

where Z}(~) is the c~-quantile of the )~}-distribution. A one-sided confidence interval is given by (

2RO* ] o,~j

In the case of one sample of ordinary order statistics from exponential distributions, the results reduce to the representations given in, e.g., Cohen (1995). For progressive type II censored order statistics we refer to Balakrishnan et al. (1999). EXAMPLE 8.3 (Sequential 3-out-of-4 system). Suppose that the s underlying systems have a common sequential 3-out-of-4 structure with a distribution function F E Y and # = 0. Without loss of generality we assume that cq = 1 and that ~2 is an arbitrary, but known positive real number. Applying the preceding results, the MLE ~9" of 0 is given by 0* = ~ sxil~-, = (4g(X!~)) + 3c~2[g(X(,2))_g(X(,~))]) in terms of the random variables X *0) X *(2) i = 1 , " " ,s. In case of an underlying i 1 i ~ ordinary k-out-of-n structure, i.e., e2 = 1, the M L E 0; of 0 is given by O°* = 2sl i~1 (49(X~~)).= + 3 [9(X~/2))_ 9(X~~))]) To analyze the different impact of the respective modeling of the system, we consider the ratio Q = O;/O*. The distribution of Q is obtained as follows. Since the random variables 4g(X!] )) and 3c~2[g(X}2)) -g(X,(]))] are iid according to F(t) = 1 - exp(-t), t _> 0, for all i = 1 , . . ,s (cf. (6.3)) we obtain

E. Cramer and U. Kamps

340

Q-

y~+~-l-

1

Ns

1

by putting

B = ~ 49(X(,]))

and

Z~ = ~

i=1

3c~2[g(X}~)) - 9(X}¢))] .

i=1

Since Y~ and Zs are independent and gamma distributed with parameters s and 1 we obtain that Zs/(Y~ + Zs) has a standard beta distribution with both parameters equal to s. Therefore the expectation of Q is given by EQ-- 1 - (1 - 1/c~2)/2 = 1 / 2 + 1/(2~a). A plot of EQ as a function of c~2 is shown in Figure 7. Let us suppose that the underlying distribution of the models is Exp(0, v~). In the sequential 3-out-of-4 model, we start with an Exp(0, 0/cq) _= Exp(0, 0)-distribution. After the first failure, the lifetime distribution of the remaining components is given by Exp(0,0/c~2). If «2 is large, the survival time of these components tends to be short. In the classical model, however, we still assume the remaining components to be Exp(0, ~)-distributed. In what follows, let us assume that the observed system is in fact a sequential 3-out-of-4 system with ~1 1 and c~2 > 0. The expectation EQ as a function of c~2 can be interpreted as follows. Both 0* and 0; estimate the lifelength of a component, namely ~, where 0* is an unbiased estimator of 0 in the considered set-up. When applying v~; as an estimator of ~, we are supposing that there is no impact of failures on remaining components. That is, the classical model, described by order statistics, is erroneously considered. The distinct values of the estimators 0; and 0* w.r.t, e2 are =

2.5-

2.0-

1.5-

EQ 1.0-

0.5-

I

I

[

I

I

I

i

2

3

4

5

6

OL2

Fig. 7. Expectation EQ

E(~;/~*) as a function of c~2.

341

Sequential k-out-@n systems

indicated by the ratio Q = 0~/0'. For illustration, a plot of the expectation of Q as a function of a z is shown in Figure 7. After the failure of the first component in our system, the hazard rate of the remaining components has changed to ~2/0. If e2 > 1, then

1(~),

~, = ~ 1 ( g + z ~ ) > ~

~+

Zs = ~ 0

a.e.,

such that 0 is underestimated by 0;. If g2 tends to infinity, E Q tends to 1/2. On the other hand, the expected lifetime of the system is estimated too optimistically if the considered system is erroneously supposed to be an ordinary 3out-of-4 system. This results from the expected lifetime of the respective systems, i.e., O

~9 ~2>10 0 < 4 -t-~=EX2,4 '

EX(*2) = 4 - ~ - ~ ~ 2

where X2,4 denotes the second order statistic in a sample of four iid random variables from Exp(0, 0).

8.2. Estimation o f the scale p a r a m e t e r in the two-parameter exponential distribution

In the sampling situation of the preceding subsection with sequential order statistics from an Exp(#, 0) distribution, the former results apply putting g(t) = t. The M L E O* of O is given by 1 ,7

ri

with X ,(°) = #, 1 < i < s, R = ~i~_1 ri (cf. Theorem 8.1). In the situation of ordinary type II censoring described by ordinary order statistics, i.e., Œ/j = 1, r i ~ r, rti = 1"t, 1 < i < s, 1 O, can be written in terms of the confluent hypergeometric function (cf. Johnson et al. 1994, p. 389). 2. va* is sufficient for va. 3. (Va*(R))R is strongly consistent for va, i.e., Va*(R) ----+ va a.e. w.r.t. R ---+oc. 4. (Va*(R))R is asymptotically normal, i.e., xffB(va*(R)/va-1) ~ JU(0, 1) w.r.t. R---+ oc. From a transformation result for MLEs it follows that va*a is the M L E of va" (see, e.g., Schervish, 1995, p. 308). This leads to the following theorem. THEOREM 8.7. The M L E O*a of the transformed parameter Oa attains the CramérRao lower bound. Since va*a is unbiased, it coincides with the U M V U E of vaa. We now focus on linear estimation of va. Linear estimates of # and va can be obtained by analogy. Suppose that (ni-j+l)c~ijTL(ni-k+l)o:ik

for

1 <j 0, m > - k / ( n - 1) - 1, Ahsanullah (1996) calculated the BLUEs as well. For completeness we give the covariance matrix of the BLUEs in an explicit form, i.e.,

Cov (OBLUE/ \ #BLUE// = 02

_1 R

s

~i=1 (ni~il) 2

s

2

~i=1 (hirn)

- ~i=1 nicql

s

R

(~i=117iO~il) k -- Ei=I ni~il

"

REMARK 9.9. As in the model of a known location parameter, particular cases are found in the literature. For references see Remark 8.4. Considering the model of record values, we have a similar simplification of the proposed estimators (for the BLUEsletR>s+I and f o r s = l l e t r l _ > 2 ) :


351

MLE:

» : minX(} ) l rg,>,>', - g(«-'9] Lt.,~/

9(Y(,°))

= 0, 1 < i < S (2),

353


s(k)

V(k) = / ~V ' n(h) ~(h) ~(~) _ p(h)Z i il i=1

The likelihood function is given by L(#,O1,O2;xij, 1 < i < s

(1), 1 < j < r}l); Yij, 1 < i < s (2), 1 < j < r } 2))

h=, ~=,

¢1of2

/ x exp

1

S(I)

I ~

(1)(1)

(1)

- Z- 2_., ni ~i~ wi

" S(2) I K-~

'Ul i=1

where

W i(1)~ W i(2)

(2)(2)

(2)

- Z- 2-_., ni ~il wi '[f2 i=1

and z are the respective realizations of W~(1),

W/(2)and Z.

THEOREM 10.1. 1. Let the common location parameter # be known. The M L E o f Oh is given by 1 vs(k) , - , (k) (h) .... (h) 0k=~2._ ni eil tvvi - # ) ,

k=1,2

.

i=1

2. Simultaneous estimation o f # , 01 and 0 2 leads to the M L E s

1 V(k) ' ~=z, Oh=~

k=l

2 .

For exponential distributions, i.e., 9 ( t ) = t, this result is shown in Cramer (2001). In the situation of part 1 of Theorem 10.1, the likelihood function obviously factorizes w.r.t. 01 and 02. Hence, the MLEs 01 and 0~ are already stated in Theorem 8.1 in a different notation. Properties are presented in Theorem 8.2. In the second part, there is no such factorization, such that a proof of the assertion is along the lines of the proof of Theorem 9.1 (cf. Cramer and Kamps, 2001). The hext theorem contains the independence of (V(1), V(2)) and Z as well as the distributions of these random variables, and hence the distributions of the above MLEs (cf. Cramer, 2001). THEOREM 10.2.

l. (V (1), V(2)) and 2. Z ~ E x p

Z are independent random variables.

#,~W+o2j

•

E. Cramerand U. Kamps

354

3. The joint distribution function of V (1) and V (2) is given by

P(V(I) ~/)1, V(2) ~ 1)2) B~(1) )~(23"~-I = ~~-1 @~-2)

["Y/(1) - 0ffvl(R(1) 1

1, 01)Fv2 (R(2), 02 )

9(2)

+-~2 F~~(R('), O1)F~~_(R (2) - 1,02)

}

,

where Fz(r, 0) = Of(r-l)! 1 fo« tr le t/Odt' z > O. The joint distribution in part 3 of Theorem 10.2 is a mixture of productgamma distributions, and V(1) and V(2) are seen to be asymptotically independent. More details can be found in Cramer (2001), e.g., the asymptotic normality

of (O~,õ2). THEOREM 10.3. Let R (1) lim

R(I),R(?)~ecR(1) -}- R(2)

-a~

(0,1),

and let limRu/R/2/~oo 7(1)/7 (2) exist and be finite. Then

~l)«:l»)

----+ ig"

((°0), ( 7

0

02/(1 -- b)

))

"

Finally we consider underlying exponential distributions, i.e., we assume g(t) = t, and we present the U M V U E of the common location parameter #. In this particular case, Theorem 10.2 includes results of Ghosh and Razmpour (1984) and Chiou and Cohen (1984) who were concerned with possibly censored samples from two exponential distributions Exp(#, 01) and Exp(#, 02). By analogy with their result it is possible to calculate the UMVUE/~ of #. The proof is based on the complete sufficiency of Z, V(1) and V(2) which is established similarly to Ghosh and Razmpour (1984) and Chiou and Cohen (1984). We arrive at the representation

(7i7V(1) - Z ) ( ~ ß ( 2 ) - « ) )=Z-

(R (1)- 1) (7@Z)V(2) -- / ) --(R(2) - 1)(~@I)V(1) - 2 )

In the case of uncensored samples (which correspond to parallel systems) and s (1) = s (2) = 1 from exponential distributions the expression reduces to the formulas of Ghosh and Razmpour (1984) and Chiou and Cohen (1984).

355


P(X
g-l(/z)

and Y~

F (2)

with F (2) (t) = 1 - exp { -

the transformed random variables namely, g(X) ~

Exp(p, 01),

and

g(t)-! , 02 ~} , t>g-l(12) _

g(X) and g(Y) are g ( Y ) ~., Exp(p,

exponentially distributed,

02),

we find

?)2

P = P ( x < r ) = P ( g ( x ) < g ( r ) ) - o, + 02

(11.1)

Tong (1974) and Bartoszewicz (1977) obtained the U M V U E of P based on independent one-parameter exponential random variables in the non-censored and censored cases, respectively. Beg (1980) considered the U M V U E of P based on distributions from exponential families with unknown scale and truncation parameters generalizing the results of Tong (1977) for the case of two unknown parameters. Constantine et al. (1986) derived a simpler expression of the UMVUE of P compared with the one in Tong (1975) based on independent random samples from gamma distributions with known, integer-valued shape parameter and unknown scale parameters. Bai and Hong (1992) considered the U M V U E of P in the exponential case with possibly different sample sizes and an unknown common location parameter (see also Cramer and Kamps, 1997b). Maximum likelihood estimation is considered in Basu (1981), Chiou and Cohen (1984) and Johnson et al. (1994). Enis and Geisser (1971) and Ghosh and


356

Sun (1997) deal with Bayesian estimation procedures. Jana (1997) compares the MLE and the U M V U E of P in the one-parameter exponential case. For further details on the estimation of P we refer to Constantine et al. (1986) and Johnson et al. (1994, p. 530/2). In what follows we are concerned with the estimation of P based on failure data from sequential k-out-of-n systems in the sampling situation of Section 10: Sequential (nlk)- r}k)+ 1)-out-of-n}k) systems are observed with known model parameters e}~), 1 _<j _ 0

(11.2)


357

(cf. Johnson et al. 1995, p. 251). In our setting, we have fll = R(2),

R(2)#l q -- R(1)va2

f12 = R(1),

Properties, moments, extensions and further applications of this distribution can be found in Libby and Novick (1982), Chen and Novick (1984) and Pham-Gia and Duong (1989). We now turn to minimum variance unbiased estimation of P in the above general sampling situation. The results in the particular case s (a) = s (2) = 1 are stated in Cramer and Kamps (1997b). In the two-sample exponential model (i.e., ~Ill) " .. = ~(1) .(23 = 1), Tong (1974) calculates the U M V U E lnO) = ~v(2) ~11 = "'" mm C~ln(2) of P for parallel systems (non-censored case), and Bartoszewicz (1977) for k-outof-n systems (censored case) (see also Johnson et al., 1994, p. 531). Let the hypergeometric function F be defined by

v(:, 8; v;z) = ~

(11.3)

where (.)j with (x)j = x(x + 1)... (x + j symbol, (x)0 = 1.

1), j _> 1, denotes Pochhammer's

THEOREM 1 1.2. The U M V U E of P is given by \

/5 =

, ,

1-Fall-R(')

~(1) ~ V(2),

,~~2~),

I'R (2)'v~2/'~

V~(1)> V~(2)

where V} ~) = V (k) + 7 (k) (Z - ~) s(k)

k = 1,2 .

= Z ,~(k)~,(k)W(k) i=1 Let R(1),R (2) > 2 and O = 01/02. Then gar j3 z

O R(2)

1

(1 + O)R/13+Æ(2/ B(R(a), R(2)) × { R(z)_I ~ R(2)_l ~ ( (1 -R(2))i(1 -R(2))/ i=0 j=o

(

(R(1))i(R(1))J

x F R (1) + R (2), 1;R (U + i + j +

R(U + i + j

1)

1;l~ß

E. Cramerand U. Kamps

358

R0) 1R0)-I ( 1 - R ( 1 ) ) i ( 1 - R ( 1 ) ) j

q- Zi=1 ~ j=l xF

-(~~)j

(

1

R(2) q_iq_j

~)} (1)

R(1)+R(2),I;R(2)+i+j+I,~TÕ

-

~Õ

REMARK 1 1.3. 1. The hypergeometric functions in the above expressions o f / 3 may be replaced by their finite series representations. We have F ( - a , 1;c;z) = ~ (-a)jzJ j=o

for a C N U {0}

(cf. Gradshteyn and Ryzhik, 1994, p. 1065) 2. Putting s (1) = s (2) = 1 and c~11~ ) = . "" ~ ~ l.(1) ù(2) ~ " ' " z ~ù(2) = 1, we obn(1) = ~11 ln(2) rain the results of Tong (1974) and Bartoszewicz (1977) (see also Johnson et al., 1994, p. 531) for non-censored and censored samples from two-parameter exponential distributions with known common location parameter, respectively. The variance of/~u in this two-sample case can be found in Cramer and Kamps (1997b). In the particular case considered by Tong (1974), Kelley et al. (1976) and Bartoszewicz (1980) derive expressions for the variance.

11.2. Estimation of P(X < Y) for an unknown common location parameter In contrast to the preceding subsection, we suppose the common location parameter to be unknown. We establish the MLE of P and its distribution, we give the U M V U E and its variance, and we state that the MLE and the U M V U E are asymptotically equivalent. THEOREM 1 1.4. The M L E of P is given by p* z

R (I) V (2)

O2

R(2) V(1) + R(1) V(2)

Öl -}- O2

with Õ~ as in Theorem 10.1, k = 1,2. The distribution of P* is a mixture of generalized three-parameter beta distributions (of. (11.2)) with density function 1 1+

Œ ah(t;R(2),R(1) - 1,11)+~'-h(t;R(2)-x->o 1,R (1) 17) '

where R(2) ~1 17 = R(I~ " zg~

and

a . .7(2) . . 01 9 (I) 02

'


359

T h e o r e m 11.1 yields that/:~ is a ratio of two independent gamma variates, and thus its distribution is a generalized three-parameter beta distribution. F r o m T h e o r e m 11.4 we deduce the following similar result. The distribution of P* is a ratio of two (dependent) r a n d o m variables, which follow a mixture of g a m m a distributions. Therefore, the distribution of P* is a mixture of generalized threeparameter beta distributions. F o r more details we refer to Cramer (2001). The U M V U E /5 of P in our setting is shown in T h e o r e m 11.5, which, for s (1) = s (2) = 1, can be found in Cramer and K a m p s (1997b). THEOREM 11.5. The U M V U E of P is given by

{

V(2)+(R(2)-1)~( 2) V(1)

. ~l~l'~_(~,~211.»l,)vl~l~

( ~ - 7 ) ~ \ - - . . . . . . . . 751

B =

(RO)-l))~(1)V(2)+(R(2)-l)9 (2)VO)

1-

(R(2)-I)9(2)v(l)+(R(1)-l)9 (t) V(2) ~,1 ,fll)) F \

........ (R0)- 1)~(1)V(2)+ (R(2) 1)9(2)V(1)

VO) < V (2)

,

_

v~Tyj

V (1) »

>

V (2)

In the particular case of different parallel systems (i.e., nll)¢; nl 2)) with s ( 1 ) = s ( 2 ) = l a n d % j .(k) = 1 for k = 1,2 and all j, Bai and H o n g (1992) derived the U M V U E of P based on underlying Exp(#,Ol) and Exp(#,O2) distributions. A corrected version is stated in Cramer and Kamps (1997a). In the particular case 9 (1) = 9 (2), the estimator/5 simplifies considerably: (/su = ) Æ =

(R(1) - 1)~(1) V(2) (R(1) - 1)p(1)V(2) q- (R(2) - 1)~(2)V(1) (R (1) _ 1)R(2)9(1)02 (R(1) - 1)R(2)9(1)01 + (R(2) -

1)R(1)~(2)~2

Although the assumption 9 (1) = 9 (2) seems to be very restrictive, it includes some important examples. First o f all, if s (1) = s (2) = s and the estimation of P is based on the observation of record values of the underlying distributions, we have 0{(1) 1 and c~}2) = ~(J~~, 1 < i < s (see Table 2). Hence the assumption is fulfilled il mm n.~rY

and thé simplified estimator can be used. Moreover, if all observed values are upper record values, the statistics W/(1) and W,.(2) are simple. We find W/i(1) = 9~/x(r}l)) (r(2)) ), such that only the largest and the smallest ,i ,) and W/(2) = g(Y,i*

observed record values in each sample are necessary to estimate P ( X < Y). A n o t h e r interesting model leading to ~(1) = ~(2) is given by progressive type II censored samples with s 0) = s (2) = s and the sample sizes N 1 , . . ,Ns. F r o m Tabie 2 we conclude that c~l~) = Ni/n i(1) and %(2) = N i / n i(2), l < i < s. Thus, for minimum variance unbiased estimation of P ( X < Y) by means of data from progressive type I1 censored samples, the simplified estimator can be applied in this situation, although the censoring schemes remain arbitrary. Since the structure of/5_ is similar to that of the M L E P*, it can be shown that its distribution is a mixture of generalized three-parameter beta distributions, too (cf. Cramer, 2001). Moreover, this mixture turns out to be a Gauss hypergeometric distribution with density function


360

(R(2) _ 1) R(2) v~(:)

R 1) + R (2) -- 2

q~(t) = \

R (I) -- 1

- - -1)R~7-1 ~- +- 1

(R~

(1 - t) R(~-2tR(~-2

×

t C (0,1) ,

{R(2)-I o~_ 1)t) R(1)+R(2)-I ' 1 + \R(~)-I where

~9=L91/0

( R (2) -

1 ) 0 1 = ( R (1) -

2

and R(I),R (2) _>2. It reduces to a beta distribution

if

1)02.

Gauss hypergeometric distributions have been introduced by Armero and Bayarri (1994, (4.1) and (4.2)) (see also Johnson et al., 1995, p. 253) as prior distributions for the traffic intensity in an M / M / 1 queue. The density function of a Gauss hypergeometric distribution with parameters ~ > 0, fi > 0, 7 and z is given by

c

X ~-1 (1 -- X)/3-1 (l+zx) ~

for0<x0

,

1 O where z = R(a)R(~-rr~_ 1 • N - 1.

This yields the variance R(2)-IK-~ (R (I) -

1

1)j

(1 +

Var/5_

+

1

-

( - 1)e(') (1 + z)R(2)Z1-R(~)-R(2)

1 +0 ×

Z)j

B(R(1) - 1,R(2)) log(1 + z ) +

(1)2 1~0

~

(-1) j

=O(R(1)' R(2)'O)'

say, i f z E ( - 1 , 0 )

,

361


Var/5_ =

(R0) - 1)(R (2) - 1) (R(I) +R(2) - 2)2(R(~) +R(2) - 1)'

ifz = 0 ,

and Var/5= = ~ (R(2), R(1), 1 )

ifz>0.

In the special case of two parallel systems with s (1) = s (2) = 1, nl 1) = nl 2) and e(k) lj = 1 for k = 1,2 and all j, t5= is given in Chiou and Cohen (1984). Assuming ~(1) = ~(2) and s0) = s (2) = 1, the variances of the U M V U E s t5 and /5 corresponding to an unknown and a known common location parameter are compared via relative efficiency in Cramer and Kamps (1997b). Since the structure of the U M V U E / 5 of P is complicated in the general situation, it is near at hand to look for asymptotic results w.r.t, an increasing number of samples. Let R = R (1) + R(2). By analogy with Lemma 1 of of Bai and Hong (1992) it is shown in Cramer (2001) that (V(k)/R(k))R«~ I is strongly consistent for 0k > 0, k = 1,2. Hence, the sequence of MLEs (P*)R is strongly consistent for arbitrary (R (1), R (2)) with min{R (1) ,R (2)} » oc. Moreover, an application of the dominated convergence theorem yields that the MLE P* is an asymptotically unbiased estimator o f P (cf. Serfling 1980, p. 11). Utilizing Theorem 11.4, i.e., the distribution of P*, we obtain the asymptotic normality of the MLE. THEOREM 11.6. Let liml~l~~,Æi2~_+o~R(1)/(R(1) T R (2)) ---- ~ E (0 71) and let the limit limR/l/,R/2/_~~ ~(1)/~(2) exist and be Jinite. Then the following assertions hold:

B((

V(1)/R (1)

01

0

and

/ p2(1_»/2.)

(11.4)

Considering the asymptotic variance in (11.4) we conclude that it is preferable to choose the sequences (R (1)) and (R (2)) such that (R (1))_~(R(2)).P)~ This choice leads to the smallest asymptotic variance of P*, i.e., 4PZ(1 By analogy with Bhattacharyya and Johnson (1974) we obtain a similar asymptotic result for the U M V U E of P.

THEOREM 11.7. L e t lime~lR:~o~R(1)/(R(2) + R (2)) = c] E (07 1) and let the limit limR/l/R/2/~~ ~(1)/~(2) exist and be finite. The U M V U E /5 and the M L E P* are asymptotically equivalent in the sense that B ( / 5 - P*) ~

0

a.e. w.r.t. R ---+ oo .

Moreover, the asymptotic distribution o f x/R(/5 - P) is that given in (11.4).

362


12. Reliability properties of sequential order statistics As pointed out in the second section, the models of order statistics and record values are contained in the model of sequential order statistics as particular cases. There is a variety of important results for order statistics and record values in connection with aging properties, such as the IFR or D F R property, and with partial orderings, such as hazard rate ordering or dispersive ordering. An excellent review on stochastic orders is given in Shaked and Shanthikumar (1994). In reliability theory, classes of distributions are considered which describe the lifelength of components or systems. There are numerous articles concerning the analysis of such families of distributions (see Barlow and Proschan, 1975, 1981; Patel, 1983; Basu, 1988). Since an ordinary order statistic Xr,n from an iid sample of random variables X 1 , . . , X ù with distribution function F represents the lifetime of an (n - r + 1)out-of-n system, one is interested in aging properties ofXr,n (more precisely, of the distribution of Xr,~), and in the transmission of aging properties. We may want to know whether the I F R property o f F implies the I F R property of the lifetime Xr,~ of the system as well as whether au aging property is transmitted from one order statistic to another. Throughout this section we assume F -1 (0+) _> 0, since the distribution function F is interpreted as lifetime distribution. If F is supposed to be absolutely continuous with density function f , then the failure rate (hazard rate) 2(.) is defined by

)~(t)- 1 f~~)(t)'__ 0 ~ F-I(o --) )(1 - F(x))(1 - F(y)) for all x,y. Concerning interpretations and examples, we refer to the literature on reliability theory (cf. Barlow and Proschan, 1975, 1981, Basu, 1988). For instance, it is well known that the following implications hold:


'IFR ~

IFRA ~

NBU'

and

'DFR ~

363

DFRA ~

NWU'

For the distributions out of the family Y (see Section 6), the I F R or D F R property is noted in Table 6. Nagaraja (1990) generalizes the above-mentioned results w.r.t, these criteria and to other neighbouring order statistics. If the rth order statistic in a sample of size n possesses the I F R (IFRA, NBU) property, so do Xr+l,n, Xr,n--~, Xr+l,n+l. Under the restriction r _< @ , the assertions remain valid for Xr+l,n+» Analogously, the D F R (DFRA, NWU) property of Xt,ù implies the corresponding property of the order statistics Xt-ic» Xr,n+l, X~-I,ù-1. Restricting to r < ~ ! , the assertions remain valid for X~-1,~-2. The above results are described and discussed in detail by Nagaraja (1990). The results may be illustrated in the triangular scheme of the order statistics (see Figures 8 and 9: If Xt0,ù0 possesses the IFR (DFR) property, then the same property holds for all order statistics belonging to the shaded area). Record values are used in models of reliability theory as well. Kochar (1990) points out that records are closely connected with occurrence times of nonhomogeneous Poisson processes and refers to surveys of Ascher and Feingold (1984) and Gupta and Kirmani (1988) (cf. Section 2.2). By analogy with transmission of the I F R property in the case of ordinary order statistics it is shown in the latter article that the I F R property of the rth record is ensured by the I F R property of the underlying distribution. Kochar (1990) generalizes this result in the sense of Takahasi (1988) using the same argument. Thus, the I F R property of some record is transmitted to the following one and the D F R property of some record is transmitted to the previous one. Moreover, if we consider a strictly increasing distribution function, it is not possible that all records possess the D F R property. Gupta and Kirmani (1988) and Kochar (1990) also consider ordering of records and record differences and present some results by analogy with results in the case of ordinary order statistics (e.g. Barlow and Proschan, 1975, p. 108). For somewhat different transmission properties of kth record values we refer to the paper of Raqab and Amin (1997).

T

nO

nO

Fig. 8. Transmission of I F R property.

364


~~° nO

nO

Fig. 9. Transmission of DFR property. For sequential order statistics only a few results without any restrictions imposed on the parameters are available up to now. In K a m p s (1994, 1995a), Pfeifer's record values and generalized order statistics are considered. The results are presented below. However, assumptions on the model parameters are made of the form m l = m2 = . rar-1 w.r.t. X2(r) , where mi = (n - 1. + 1)~i - (n - 1)~i+1 - 1, l 0, which here are denoted by X,(I'~),.. ,X,(,k'k) in order to indicate the sample size. 1. L e t r i) ,

(11) we also readily have the property that Cov(Xi:m:n,Xj:m:n) = Var(Xi:m:n)

for j > i .

(12)

These results will enable us to develop exact inference for the parameters of both one- and two-parameter exponential distributions based on progressively Type-II right censored samples. This will be explored further in Section 4. 3.1.2. The uniform distribution

Let us move on now to the Uniform(0,1) distribution with probability density function f(u)=

1,

O

-1,

B(mR~;~»Rm1 '0)(k+') = (k + 1)#m:m:n,... (R~R,ù 1,0)(k) + (n - R 1 X

-R2

.....

Rm-I - m + 1)

B(Rl,...,Rm 2,Rm-l+l) (k+l) m-l:m-l:n

1-P A ~ ~ - i~, ~n -- 2)

rp~+l X L i

. (~~, ,~,ù-~s,,-l;~+l~l --]Xm

l:m l : n - 1

J

"

(57)

The recurrence relations presented in this section are complete in the sense that they will enable one to compute all the single moments of all progressively Type-II right censored order statistics from doubly, right (Q--+ 0), and left (P--+ 1) truncated exponential distributions for all sample sizes and all censoring schemes. Also, if we let P --+ 1 and Q ---+0, these results reduce to the recurrence relations for single moments of progressively Type-II right censoredorder statistics from the standard exponential distribution (see Aggarwala and Balakrishnan, 1996).

R. Aggarwala

392

3.3.2. Recurrence relations f o r p r o d u c t m o m e n t s

Again exploiting the characterizing differential equation for the doubly truncated exponential distribution, we arrive at the following recurrence relations for product moment of progressively Type-II right censored order statistics. THEOREM 14. For 1 < i < j i+l

We can therefore obtain the exact B L U E of 0 and its variance, using (66) and (67). Explicit expressions of the BLUE and its variance are as follows: 1 m

02

O* = m~-~'(Ri~=l @ 1)~i. . . . It can be shown that freedom.

2mO*/O is

and

Var(0*) = --m

(72)

distributed as chi-square with 2m degrees of

398

R. Aggarwala

REMARK 18. It is of interest to observe that the precision of the BLUE of in (72) depends only on m and n, and not on the progressive censoring scheme

(R1,..

,Rm).

REMARK 19. Balakrishnan et al. (1999b) have considered estimation of the scale parameter 0 based on k progressively censored samples of varying sizes and censoring schemes. 4.1.1.2. The Pareto Distribution For a progressively Type-II right censored sample from the standard Pareto(v) distribution, let us denote the corresponding progressively Type-Il right censored order statistics by Xl:m:~, X 2 . . . . . ' ' ' , X m : m m . Now, let us represent the progressively Type-II right censored order statistics with the same censoring scheine from the Pareto (v, 0) distribution, that is, with probability density function f(y;v,O)=õ

,

y>0,

v>0,

0>0

(73)

by Y/:m:~, i = 1 , 2 , . . , m . Using the expressions for the means, variances, and covariances of X1..... X2:m:~,.. ,Xm .... obtained in Section 3.1.3, we see that the m x m variance-covariance matrix is again of the special form «ij = sitj which can be inverted explicitly using (71). The elements of the symmetrie tri-diagonal inverted variance-covariance matrix of X, (ci,J), are given by ~)iTi+l __ f12i f12i+~

ci,i =

i = 1,2,..,

m -- 1,

(I~~=i ~k)(])i- f12)(~)i÷1- fl2÷i) ' 1

C m~m

fii+l

ci,i--1 = i

i = 1 2,..,

m -- 1,

f12

.

c ''J = 0 otherwise ,

(74)

where cq, fii, and Yi, i = 1 , 2 , . . , m, are as given in notation (32) in Section 3.1.3. 4.1.1.3. First-order approximation to the BLUE Let us again assume that the progressively Type-II right censored sample y(R1,..,Rù,) l:m:n ~ v(RI,...,I~~) *2:m:n

V( 0 ,

0>0

as the t i m e - t o - b r e a k d o w n distribution. Using the exact results from (72), we find that the B L U E of 0 is 1

m

-- ~-~,(Rg + 1)yg. . . . . m ~.

=

9.110

with a standard error of 0 * / v ~ = 3.221.

Table 1 Progressively censored sample generated from the times to breakdown data for insulating fluid tested at 34 KV by Nelson (1982) i

1

2

3

4

5

6

yi:8:19

0.19

0.78

Ri

0

0

7

8

0.96

1.31

2.78

3

0

3

4.85

6.50

7.35

0

0

5

Progressivecensoring."A review

401

Using the first-order a p p r o x i m a t i o n to the B L U E of 0, we note that the standardized variable X = Y/O has a standard exponential distribution in which case

F(x)=l-e

x, F - l ( u ) = _ l n ( l _ u )

and

F l/'/(u)=l/(1-u)

.

F r o m (77), we determine the B L U E of 0 to be a p p r o x i m a t e l y 0* = (0.12305 x 0.19) + (0.12305 × 0.78) + (0.47255 x 0.96) + (0.12458 × 1.31) -4- (0.47153 × 2.78) -4- (0.12798 x 4.85) + (0.12808 × 6.50) + (0.82641 × 7.35) = 9.57 . F r o m (79) we have Var(0*) = 0.11713 02 so that we obtain the standard error of the estimate 0* to be

SE(O*) = 0"(0.11713) 1/2 = 3.28 . These exact and a p p r o x i m a t e values are very close, with n = 19 and m = 8.

4.1.2. Two-parameter mode& Consider again an arbitrary "'standard" continuous distribution F(x). Suppose n o w that we believe our progressively censored observations to be represented by the linear t r a n s f o r m a t i o n Y =/~1 + eX, where the vector X represents a vector of m progressively censored order statistics f r o m the standard distribution F(x); then the best linear unbiased estimators of/~ and « will be obtained by minimizing the generalized variance Q(0) = ( Y - A0)'12 -1 ( Y - A0) with respect to 0 where B = (#, «)', A is the m x 2 matrix (1, la), 1 is the m × 1 vector with c o m p o n e n t s all l's, and la is the m e a n vector of X and 12 is the v a r i a n c e - c o v a r i a n c e matrix of X. The m i n i m u m occurs when #*=-laTY

and

cr*=lTY

,

(80)

where F = 12-1(1la/- lal')12 1lA and A : (1'12 11) (laQ2-1la) -- (lt•-lla) 2. F r o m these expressions, variances and the covariance of the estimators are readily obtained as

a21a'12-11a/A, Var(«*) = 0-21112-il/A, Var(#*) =

Cov(~*, «*) = _«21a,~-i 1/~ .

(81)

Alternate explicit expressions for the B L U E s and their variances and covariance are as follows: m

j=l

m

i--1

m

m

l--1 k - 1

R. Aggarwala

402

(7*

«'i k (#1 - #~)cldYj:m:. ,

=

(83)

j = l i=l l=l k=l

where

(

/(,~~#,#jcz'j)(- ~i~=ip,c")'

A= ~-~.~ci'i i=1 j=l

"

,/ \ i = 1 j = l

(84)

i=1

and c'« are the elements of the inverted variance~ovariance matrix of X. Furthermore, V a r ( Æ * ) = r 7 2 ( ~ "i=1 ~ ~ ~1=1 --~~#ißjci'J)/A'\

(85)

Var('*)='=('~~=1~_ici'Y)/A

(86)

and (87)

4.1.2.1. The exponential distribution Consider the two-parameter exponential distribution with probability density function

f(y; ~, «) =-le-(y-")/«,

y > ~,

o-

« > 0.

(8a)

Let us denote the progressively Type-II right censored sample from this distribution by Y~:m:~, i = 1 , 2 , . . ,m. Then, again using the moments for the standard exponential distribution obtained in Section 3.1.1 and the formulas in (80), we derive, after rauch algebraic simplification, the BLUEs of g and a to be (see Viveros and Balakrishnan, 1994; Balakrishnan and Aggarwala, 2000) 1

P* = Y1....

m

n(m-- 1) +E i( :R2z

1)(~:m:n

--

Yl:m:n)

(89)

and

«,

m

_1

~( 1 = Ri @

1)(Y[ .... - Yl:m:n) ,

(90)

respectively. Further, upon using (81), the variances and covariance of these estimators are given by

Progressive censoring." A review

403

mo- 2

Var(#*) - n 2 ( m

- 1)

(91)

O-2

Var(a*)

-

(92)

m--1

and O-2

n(m-

Cov(#*,o-*) -

1)

(92)

We can show that 2(m - 1)o-*/o- has a chi-square distribution with 2(m - 1) degrees of freedom. This fact can then be used to develop confidence intervals or tests of hypothesis about o-. Using the fact that 2(Yl:m:n - #)/o-* has a chi-square distribution with 2 degrees of freedom and that the random variables 2 ( m - 1)o-*/o- and 2(Y1.... - - # ) / o - * are statistically independent, we find that u ) u t- ~ has an F-distribution with (2,2m - 2) degrees of freedom. This pivotal quantity can be used to perform tests of inference concerning #; see Viveros and Balakrishnan (1994). REMARK 20. It is important to note here that the precision of the BLUEs of # and o-, for the exponential distribution depend only on m, and n, and not on the progressive censoring scheine (R1,..,Rm). REMARK 21. A number of results have also been established for the more general case of left and right progressive censoring, or general progressive TypeII censoring. They are presented, for example, in Balakrishnan and Aggarwala (2000). 4.1.1.2. The Pareto distribution Let us denote the progressively Type-II right censored order statistics from the location-scale shifted Pareto (v) distribution, that is, with density function

f(y;v,#,a)

= - v (~~ff_) (7

~ ~, y > # + o - ,

v>0,

-oo0 . (103)

Denoting the observed (ordered) progressively censored sample by yi, i = 1 , . . , k, the log-likelihood function to be maximized for the location and scale parameters will be

lnL(p,a) = c o n s t a n t - n l n a - ~

l k~ß'~{Yi-- p'~2

i2__~_l~ , ~ )

~-~'~-

+2_.,t¢i

"=

ln(1-F/)

i=1

(104) where B-p (9"

and q~(.) is the cumulative distribution function of the standard normal (p = 0, « = 1) distribution. To maximize the log-likelihood, the score equations to be solved are

2ki-1Yi

k

2~='(Y'-Y/2 k

m Ri

= Y = P - « ~i=17 ; zi' «2 1 - ~ ~ ~ ~ z , - ~ i~l

zi

(10»)

i=l

where Zi = (p(~i)/1 - 4~(~i) and ~0(.) is the probability density function of the standard normal distribution. Cohen (1963) has discussed the numerical solutions

R. Aggarwala

410

of these equations in detail. Now, many mathematical computer packages contain algorithms to solve the above system of equations efficiently. Note that maximum likelihood estimation for progressively Type-II right censored samples will result in the same system of equations, with the censoring times, T~, replaced by the failure times, yi. REMARK 22. In determining the variances and covariance of the MLEs, one could examine (at least asymptotically) the asymptotic variance-covariance matrix,

(106)

4.4.2. The exponential distribution

We have already addressed best linear unbiased estimation for the one- and twoparameter exponential distributions. Let us denote the observed progressively Type-II right censored order statistics from the one-parameter exponential distribution by yi, i = 1 , 2 , . . . , m. The M L E of the scale parameter 0 is then given by =--

nyl +

/11

+ 1)(yj-Yl

=

j=l

+ 1)yj ,

(107)

"=

which becomes identical with the BLUE of 0 given in (72). Cohen (1963) derived the MLE for 0 when the observed sample is a progressively Type-I right censored sample. The estimator is k + K-,m R.T. = ~i=1Yi Total Accumulated Life ~i< "' = k k '

(108)

and it reduces to (107) when we adjust it for Type-II samples, that is T, are replaced by yi, and k, the number of observed failures times, is replaced by m. Let us now denote the progressively Type-II right censored order statistics from the two-parameter exponential distribution (location parameter /z, scale parameter a) by Yi, i = 1 , . . . ,m. The log-likelihood function based on this progressively Type-II right censored sample is given by m

lnL = ( C o n s t a n t ) - m l n a

-

Z(Ri+1)( i 1

y/:m:n- \

~) ,

(109)

O-

which is monotonically increasing in # so that the MLE of # is ~ = yl. In this case, the M L E of o- is given by 1

m

G = m~'(Ri~=2 q-

1)(y/-Yl)

•

(110)

Progressive censoring: A review

411

It may be noted that the MLEs ~ and ~ are both biased, and in fact, they are simply the BLUEs adjusted for their bias. Therefore, the pivotal quantities established in Section 4.1.2 for inference using BLUEs will also hold for MLEs. See Viveros and Balakrishnan (1994). EXAMPLE 5. Earlier in Example 1, we considered Nelson's data on times to breakdown, where a one-parameter exponential model seemed appropriate. The one-parameter B L U E in that case was found to be 0* = 9.110 with aA standard error of 3.221. Using (107), the M L E of 0 is found numerically to be 0 = 9.111. 4.4.3. The Weibull distribution In one of a series of papers on likelihood estimation under progressive censoring that appeared in Technometrics, Cohen (1975) considered both maximum likelihood and "modified maximum likelihood" estimation for the three-parameter [location(/z) scale(a) - shape(0)] Weibull distribution with cumulative distribution function F(y)=l-e

(y_~)o/«

y>/~,

-cxD0.

(111)

The log-likelihood function for a progressively Type-I right censored sample from this distribution is given by k

in L = (Constant) + k in 0 - k in a + (0 - 1) Z

i n ( y / - #)

i=1

(Yi - #)o+ O-

Ri(Ti - #)o

,

(112)

i=1

where y~ is the observed (ordered) progressively censored sample. Again, the results which we will present for progressively Type-I right censored samples may be adapted to Type-II scenarios and simplified in the usual way. From inspection of the likelihood function, we see that we may consider the three cases 0 < 1, 0 = 1, and 0 > 1 separately. If it is known that Õ = 1, then we have a simple location-scale exponential distribution, whose estimators we have already derived in the previous subsection. If 0 < 1, then we see that the likelihood function becomes infinite as # ~ yl, and therefore ~ = yl [in fact, as Cohen (1975) has pointed out, ~ = yl - 5, where q is the unit ofprecision of measurements made.] The additional score equations to be solved for a and 0 in this case are given by

~17 -- 0 = ----Œ-~~

(Yi --

/2)0@

ZRi(Tii=l

- #)0

,

(113)

OlnL k k ~Õ- - 0 = ö + ~ ln(yi - ,u) i=1

-

(yi -

~)°ln(yi

-

#) +

/~,(r~

-

~,)°ln(r,

-

~)

(114)

R. Aggarwala

412

We can now eliminate « between these two equations, obtaining a single equation as Y'~~/k=l(Yi - #)°ln(» - #) + ~i%a Ri(Ti -/~)°ln(T/-/~) k ~ i = 1 (Yi -- ~)0[_ ~ m l R i ( T i _ #)0

1 ö

1 k lc/__~ll n ( y i - / ~ ) = 0 '

(115)

When # or its estimate is known, as in the present case and the two-parameter case discussed by Cohen (1966), one needs to solve only this equation for 0 and then obtain the estimate for a by substitution back into the original equation, lnL/aa = 0, given above. Finally, for the case when 0 > 1, one taust solve a system of three equations, consisting of the two equations ~lnL/êa = 0 and ôlnL/aO = 0 given above, along with the equation

~~n~ 0E~ -

o =

-

(y, _

~)o-1+

O"

~

Ri(r~

-

~~01

i=l k

- (0 - 1 ) Z ( y i - #)-1

(116)

i=1

Again, simplification by eliminating the parameter a is possible. Cohen (1975) has discussed the numerical solutions to these equations in more detail, using yl as an initial approximation for 3. Many available computer algorithms will have suitable methods for solving this system of equations directly. The reader is referred to Balakrishnan and Aggarwala (2000) for further discussion on maximum likelihood estimation. A number of interesting results have been obtained, including results for Laplace and Pareto parameters, as well as consideration of MLEs under left and right (or general) progressive censoring schemes. Other topics in inference, including linear prediction and conditional inference, are also presented there.

5. Related topics in progressive eensoring

5.1. Simulation Many of the studies carried out on progressive censoring are computation in nature. Balakrishnan and Sandhu (1995) presented the following simple algorithm based on the mathematical properties of progressively censored order statistics from uniform distributions, for simulation of progressively censored order statistics from arbitrary continuous distributions. 1. Generate m independent Uniform (0, 1) observations W1, W 2 , . . , 1/ i+~j=m i+lRJ 2. Set Vi = Wz. for i = 1 , 2 , . . , m.

Wm.

413


3. Set Ui:m:n = 1 -

gingm 1"'" Vm i+1 for

i = 1,2,..»m. T h e n

U1 .....

U2:m:n,...,

Um.... is the required progressively Type-II right censored sample from the Uniform (0, 1) distribution. 4. Finally, we set X~:m:n = F-I(Ui .... ), for i = 1 , 2 , . . ,m, where F - l ( . ) is the inverse cumulative distribution function of the lifetime distribution under consideration. Then XI ...... X2:m:n,... ,Xm:m:, is the required progressively Type-II right censored sample from the distribution F(.). The above simulation algorithm requires exactly m pseudo-random uniform observations, and does not require any sorting. Similar algorithms can be established using the mathematical properties we have presented in Section 3 for exponential and Pareto distributions.

5.2. Optimal censoring schemes The question of choosing optimal values of R I , R 2 , . . ,Rm when considering a progressive Type-II right censoring scheine is an important one to consider from a practical point of view, and as it turns out, it also gives rise to a number of interesting problems, in the areas of optimization, numerical analysis, simulation and programming. The first question to consider is out definition of optimality. That is, what criterion will be used to determine whether one scheine is better than another? In mathematical terms, we taust specify an objective function to be optimized before we can determine the optimal progressive censoring scheine to be employed, as different censoring schemes may perform optimally under different objective functions. First, ler us consider a practitioner who is interested in designing a warranty. This individual will be interested in the estimation of the time after which, say 95 per cent of the product will survive, that is, the manufacturer of the product is willing to repair 5% of all items produced. In this case, we are interested in estimating the fifth percentile of the product's failure-time distribution. This being a single (univariate) estimator, we could define the best scheme as the scheine which minimizes the mean squared error of the estimator. That i~ the objective function (to be minimized in this case) would be MSE(~0.05)where 40.05 represents the estimator for the fifth percentile, the parameter of interest. This has been discussed by Aggarwala (1998). A similar objective function may be defined for any case where a single parameter is to be estimated. We have discussed two-parameter (for example, location-scale) estimation extensively in Section 4. In the case where more than one parameter is to be estimated, there is a variance-covariance structure of the estimators which one may wish to consider. In the case of two-parameter estimation, optimality may be defined in terms of the trace or determinant of the variance-covariance matrix of the estimators, which makes sense particularly if the estimators are unbiased. In the case of biased estimators, we may want to replace the variance terms in out objective functions by the corresponding mean squared errors. More complicated objective functions can also be considered. For example, suppose the cost of experimentation is of concern, and will be proportional to the

R. Aggarwala

414

time on test of the experiment. Then one may wish to add to the objective function terms containing

E(Y~m(RI....

'R2""'Rm)

) °r P(Y~:%'::2,'R~) >

TU)

,

(1 17)

where y(R~,e2,..,e~) is the last observed failure time in the experiment and Tv is a m:m:n time by which we would like to end experimentation with high probability. Similarly, one may want to include in the objective function terms involving the variance of ~vm(e~,e2,'',R'n) or may have an interest in recycling live items removed, :m:n » and therefore add a term such as m

Zconstanti ×

Ri ,

(118)

i--1

where, perhaps the constants decrease with i. It is clear that there is an infinite number of possibilities, and the practitioner is able to customize the objective function to be optimized according to his or her priorities. Once the objective function has been satisfactorily defined based on n units available to put on test, the next question to consider is the one of actually finding the optimal censoring scheine. Suppose that a practitioner would like to observe m complete failure times, and taust determine what the best scheine to use in this case is. In this finite sample situation, one may list all (m-a) ù-a possible choices of censoring schemes and corresponding values of objective functions, and determine the best value (i.e. the value which optimizes the objective function) or a region of satisfactory values from this list, and either pick the best scheine or one which gives a value very close to the best but which may be practically more convenient. Alternatively, there may be cases where a simple closed-form expression of the objective function is available, and optimization is somewhat more mathematically elegant. Notice that m and n are chosen in advance here. If one or both of these are also to be determined, one may proceed by deciding upon values of m and n which are feasible given an agreeable value of the objective function. Aggarwala (1996) and Balakrishnan and Aggarwala (2000) have considered both the trace and the determinant of the varianc~covariance matrix of BLUEs for two-parameter location/scale distributions as objective functions of interest. In Sections 3 and 4, we discussed various ways of calculating these values, and as such, the actual value of the objective function may be determined, either in closed form or through a computer algorithm, for any scheme and for a number of distributions. The feasibility of computation is an important part of defining an objective function. Balakrishnan and Aggarwala (2000) have presented tables of optimal censoring schemes for these two objective functions for the extreme value, negative extreme value, log-normal and normal distributions which were obtained by calculating the objective function based on all (m-l) n-1 censoring schemes and choosing the best and worst values. They also note that for the exponential distribution, we have already seen that the variance-covariance matrix of BLUEs is independent of the censoring scheine chosen, and therefore,

Progressive censoring."A review

415

for these objective functions, the censoring scheme may be chosen for the sake of practicality and convenience. Two objective functions may be compared to one another through their relative efficiency, which can be defined as Objective function (Scheme B) . Objective function (Scheme A) '

(119)

hence, if one is interested in determining a region of satisfactory schemes, she or he can define that region in terms of efficiencies. For example, an experimenter may be satisfied with choosing any scheme which is at least 98% as efficient as the optimal scheme. For the distributions and values of n and m considered in Balakrishnan and Aggarwala (2000), the conventional Type-II right censoring scheme is orten the least efficient in terms of the objective functions considered. This was seen to be true for the extreme value and normal distributions, whereas for the log-normal distribution, the conventional scheme was the most efficient. This highlights the importance of having some idea of the distribution of lifetime failures. Goodness of fit algorithms for progressively censored samples is an area which is currently being investigated. As an example, the results for trace and determinant optimal and least optimal (where the matrix of interest is the variance-covariance matrix of location-scale BLUEs) progressive Type-II right censoring schemes for the two parameter normal distribution are given in Tables 3-5 for selected values of m and n. The reader is referred to Balakrishnan and Aggarwala (2000) for a more indepth discussion of such tables. Note that since an optimal scheme in this case would minimize the objective function, large values for efficiencies indicate that Scheme A is more efficient than Scheme B (which is taken to be the conventional Type-II right censoring scheme in Tables 3 and 4). Table 3 gives efficiencies (line 1) and traces (line 2) of the optimal scheme where efficiencies are with respect to the conventional censoring scheme. Table 4 gives the same for determinants. Table 5 provides coefficients, variances and the covariance of BLUEs for selected schemes. In that table, the following notations are used beside censoring schemes: (Note: conventional censoring schemes are always included, regardless of whether or not they fall into one of the following categories): + = trace optimal, * = determinant optimal, - = least optimal (trace), / = least optimal (determinant).

5.3. Acceptance sampling plans Acceptance sampling plans are perhaps the oldest and most extensively used statistical tool in industry. Although there is currently a strong shift towards product improvement (versus product monitoring), acceptance sampling based on inspection of finished products will continue to be a component of most company-wide quality initiatives. One form of product inspection is through life

R. Aggarwala

416

C',l

Cq

Cq t-q

e4

0

© "~

oc

ù~

'~ ©

ù~

ee3 ~D

[.,«

t--


z

0 ù~

c'--

©

.L

417

R. Aggarwala

418

T"

b ,> o

U"

;>

;>

© ",~

0

~D

+.a

i

o © I I I I I I I I I I

[

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l

l

Il

..= .o

0

.o

o&

~ù~

r)

~~~~~~°°°~

t~

c,..)

+~ ~ ~ ~

~ + ~ ° ~ d ~ ~ ~ ~ O ~ & ~ ~ ~ °

&~ ~ ~


I

I

I

I

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

l

419

l

[

l

l

l

420

R. Aggarwala

testing experimentation (sometimes referred to as reliability or life-test [acceptance] sampling plans). Naturally, censored samples will be of interest in this context (sometimes described as failure-censored reliability [acceptance] sampling plans). This discussion will focus upon the construction of failure-censored reliability acceptance sampling plans based on progressively Type-Il right censored samples. It should be noted that these tools need not be applied only to finished products, but may be modified and implemented at earlier points in production. Owen (1964, 1969) discussed acceptance sampling plans for normal and nonnormal distributions, while Hosona and Kase (1981) considered sampling plans based on the double exponential distribution. Fertig and Mann (1980) developed reliability sampling plans for Weibull and extreme value distributions, Schneider (1989) discussed failure-censored reliability sampling plans for Weibull and lognormal distributions, and Balasooriya (1995) discussed failure-censored reliability sampling plans for the exponential distribution. Kocherlakota and Balakrishnan (1984, 1986) proposed robust acceptance sampling plans based on exponential distributions. With regard to progressively Type-Il right censored samples, Balasooriya and Saw (1998) have developed reliability sampling plans and operating characteristic curves for two-parameter exponential distributions, and Balasooriya and Balakrishnan (2000) have developed reliability sampling plans for the log-normal distribution based on progressively Type-II right censored samples. These are further addressed in Balakrishnan and Aggarwala (2000). Many of these references provide examples using MIL STD standards for producer and consumer risks. For the purpose of developing such acceptance sampling plans, we may either use exact (ML/BLU) estimators of the parameters and their exact distributional properties or we may use first-order approximate estimators of the parameters and their distributional properties. We will consider the exponential distribution for illustration using exact results. Balakrishnan and Aggarwala (2000) also address the log-normal lifetime distribution as an illustration using approximate results. Thus, we will consider the two-parameter exponential distribution with probability density function f(y; #, «) = l_ e_0~_~)/«, y>#,_ « > 0 cr

,

(120)

where # is the location parameter and cr is the scale parameter. In the present context, # is the guarantee period and a is the mean residual lifetime, the mean lifetime being # + a. 5.3.1. One-sided sampling plans

For one-sided acceptance sampling plans, based on the progressively Type-II (1 R~), right censored sample Yl:m:~"' ~m:m:nV(Rl'""Rm) (denoted from now on by 111...... . . . , Ym..... for convenience) observed from a lot of n units, the lot is accepted if either q51(Y1..... . . . , Ym.... ) _> L or ~b2(Yl:m:ù,.. , Ym.... ) _< U where the functions q51 and q52 are statistics based on the observed sample, determined so •

•

,

Progressive censoring." A review

421

that in each case certain probability statements are satisfied. Depending on whether a right-sided or left-sided acceptance sampling plan is desired, one of the above two inequalities will be used. Suppose that L and U (often referred to as the lower specification and upper specification limits, and often determined based on much larger collections of data over time) represent quantiles from the two-parameter exponential lifetime distribution. Let Kp = - l o g ( 1 - p ) and K~_p = - l o g p denote the lower and upper p-percentage points of the standard exponential distribution, respectively. Then, from the equations Pr(YU)=p

,

(121)

we have L-#

Kp-

and

U-#

Kl-p---

er

(122)

er

The one-sided acceptance sampling plans we will discuss here will coincide with one-sided tolerance limits (see Engelhardt and Bain, 1978 for discussion based on complete samples), as well as one-sided eonfidence intervals for quantiles based on progressively Type-II right censored samples, diseussed, for example, by Aggarwala (1998) for extreme value lifetime distributions. 5.3.1.1. Case 1:/~ unknown and er known It may be the case that the location parameter # is highly variable, whereas the location parameter a is very stable, and therefore considered a known, constant quantity. In these cases, we will accept the lot of size n if either/z + kla >_L or /2 + k2er _< U, depending upon whether we are interested in a right-sided or leftsided acceptance rule. Here,/~ = Yl:m:nis the M L E of/z, and kl and k2 need to be determined so that

Pr(fi+k~er>_L)=l- 7 and

Pr(/)+k2er_ O,

i = 1, . . , m

i = 1, . . , m

(15)

,

(16)

.

Using the above expression, we approximate the likelihood Eqs. (7) and (8) by ô log L mzr z~ m ô# ~ - ~ + äß~-'(Ri,y..{_ log L

- - ~

~O"

m O"

7c

m

+ 2){7i + 6iYi} = 0 ,

7"C

Zyi+~~-~~(Ri+

0"~/3 /=1

(17)

m

oV3 i=1

2)yi{?i+6iYi}=O

.

(18)

N. Balakrishnan and N. Kannan

436

Eq. (17) m a y n o w be rewritten as m

m

--mq- Z ( R i 4 -

2)Yi q- ~-~~(Riq- 2 ) ~ i ( ~ )

i=1

(19)

=0

i=1

which yields the estimator of # as

B = ~i~_1 (Ri + 2)6i =K+La

Ri + 2)(Sixi + cr

Ri + 2)7i - m

,

(20)

where

K = ~i~=] (Ri + 2)aixi

(21)

m R i + 2)& 2,=1(

L = ~im] (Ri + 2)7i - m ~ i =m1 ( R i + 2)6i

(22)

Eq. (18) m a y n o w be rewritten as m

G

m

GX/3~=I \ 7~

O" ,I +

m

_

( X ~ )

Riq-2)Yi(T)

2

+~-~i~.l(Ri+2)6i_

=0

.

(23)

Replacing # n o w by K +LG as in (20), we have from (18) m

m

t-

Ri + 2)7i - 1}(xi - K) +

0-

Ri + 2)6i(x~ - K) 2

_

mL~

Lrc

2Lrc

~

m~

L2~ y_. m

a 7 ,__Ll/~'+ 2)~~+ ; 7 ~ ( ~ ,

«,~

+ 2)ô,

"'

Z(Rii-1

(24)

-~ 2)~)i(xi -- K) = 0 .

The last four terms can be shown to vanish, leaving us with the quadratic equation m

mG2 .

.

m

R i. + 2)yi.

1}(x/

K)

Ri + 2)6i(xi

-

K) 2 =

0

(25)

Point and interval estimation for parameters of the logistic distribution

437

or

ma 2 - A l a

-A2 = 0 ,

where 7E

m

A1 : ~ Z { ( R i

+ 2)7i - 1}(xi - K),

i=1

7~

m

A 2 : ~Z(Ri+

2)Õi(xi-K) 2 > 0 .

i=1

Eq. (25) is a quadratic equation in a, with the roots given by

ä=

A1 i vIA 2 + 4mA2

2m Since A2 > 0, only one root is admissible, and hence the approximate MLE of a is given by

A1 + ~/A~ + 4mA2 ä=

(26)

2m

The approximate MLEs are thus explicitly given by the expressions in (20) and (26). This procedure allows us to obtain estimators for # and a explicitly. We need to evaluate the performance of these approximate estimators by comparing their efficiency with those of the MLEs obtained by solving the likelihood equations numerically. These approximate solutions may provide us with an excellent starting value for the iterative solution of the likelihood Eqs. (7) and (8).

4. Weighted least squares estimation

Consider the cdf of the L(#, a 2) given by 1

F(x; #, a) = 1 + e-~(x ~)/o4g

(27)

Since F(Xi:m:n) ~ U/:m:n as mentioned earlier, we have

E(Xi:m:n) =F

v ~ 1 /Il C C~i:m:n) l(~i:m:n)=#--0"~n~- e~.... / '

where C~i:m:~is as given in (11). We can construct weighted least squares estimators of # and a using the observed data. The BLUE of the parameter vector 0T = (#, a) is given by

N. Balakr~hnan and N. Kannan

438

0L = ( w T ~ - l W )

-1WT~-IX

,

(28)

where X is the m x 1 vector containing the observed progressively Type-II censored data, W is the m x 2 matrix with the first column containing all ls and the second column containing the terms -(x/J/~)ln((1 -C~i..m:~)/C~i.... ), and ~2 is the variance-covariance matrix of the observed progressively censored order statistics which may once again be approximated using the moments of the uniform progressively censored order statistics. The first-order approximations for these quantities have been presented by Balakrishnan and Aggarwala (2000). The BLUEs are known to be asymptotically unbiased and efficient for large m. Furthermore, these estimators should also provide good estimates for large effective sample sizes, i.e. large m, and may therefore provide a good starting value to begin the numerical iterations required for the determination of the MLEs. It is of interest to mention here that upon using this approximate expression of Z and the resulting symmetric tri-diagonal matrix as its inverse, Balakfishnan and Rao (1997) have derived explicit expressions for the BLUEs of # and a in (28) and the variances and covariance of these estimators.

5. O b s e r v e d

and expected

Fisher information

In this section, we compute the observed and expected Fisher information based on the likelihood as well as the approximate likelihood equations. These will enable us to develop pivotal quantities based on the limiting normal distribution and then examine the probability coverages of these pivotal quantities through Monte Carlo simulations. We now derive the observed Fisher information for the likelihood Eqs. (7) and (8). We find 62 in L 6# 2 62 i n

L

Ô#6o-

~z m ~rg--/wZ(Ri-v»

rmr

--

+ 2)f(yi)

(29)

,

i=1

o'2 V/3

n

m

7"2

2-X/~ i~_l(Ri-2)F(yi) o" _

m

a B v / ~ i=1 Z ( R i + 2)yif(Yi) ,

(30) Õ2 in

L

_

m

2~

m

2~

m

m

From these expressions, we now compute the expected Fisher information. First of all, we have

Point and interval estimation for parameters of the logistic distribution

( ô21nq_ E

~1~2 J

439

~ m «SV~i~=l (Ri + 2)E[f(Yi)]

where E[f(Y~)] = ~ E [ F ( Y ~ ) { 1 - F(Y~-)}] = ~ 3 E[{1 - F ( Y i ) } ] - ~ 3 E [ { 1 - F(Y,-)} 2]

D

F r o m the joint density of Y1,. •, Ym, we readily find

E[{1 - F(y.)}] = A ( n , m

-

1) f--. f f(Yl){ 1 - F(y,)} fr, a* > 0 ,

(2.5)

where «* = a/n (see Balasooriya, 1995). Thus, one could also use the above sampling plans with straightforward modifications when specimens are to be tested in assemblies of fixed size, and failure of an item is considered as an assembly failure.

2.2. The Weibull (extreme-value) distribution Fertig and Mann (1980) discussed life-test sampling plans for the Weibull distribution (and extreme-value distribution). For Type II censoring, failurecensored variables-sampling plans for lognormal and Weibull distributions were studied by Schneider (1989). Variables-sampling plans under Type II progressive censoring were considered by Balasooriya et al. (2000). Note that if random variable Z has a Weibull distribution with pdf

f(y;c~,fl)

C~

;3-1exp -

,

y > 0 ,

(2.6)

where c~and/~ are, respectively, the scale and shape parameters of the distribution, then X = in Z has an extreme-value distribution with location parameter # = in and scale parameters cr = 1/il. Thus the function 9(.) in (2.1) is given by

9(u) = euexp{-e"},

- o o < u < oo .

(2.7)

I f / i and ~ denote the MLEs, then using the large sample theory

B -- ke ~ N(}t - ko-, [711 (Y/) -~- k2722(n) - 2k712(n)]a2/n)

,

where 7ij(n)'s are the variance and covariance factors in the asymptotic covariance matrix of (/1, ä)' given by (in terms of Fisher information matrix)

F(n)

[y1](~ )

y12(n) 1

0.2, [«ll(n ) «12(n)]-1 Lv21(n) v22(n)J = n LJ21(n) J22(r/) J

'

463

Progressively censored variables-sampling plans f o r life testing

where 0-2 //8 2 in 5o'~ " - / " / = 7 ~ ~ w s[- ù . ) - /

x-'m S.e~* ù ,

(3"2 / ' ~ 2 l n ~ ~

J12(ù/= J21(ù/= 7 E t ~ T õ 7 ) ~ 0-'

/ ~ 2 In y ' ~

~m=l&eu'(1 + # i )

ù

q,

~,'."-, {13Æ(~2o + 2~, - ~~]S~e", - ~ 3 n

and q = min. The likelihood function £0, given a Type II progressively censored sample Zl,n,..-,Zm,n (with corresponding censoring numbers R 1 , . . , R m ) from Weibull distribution in (2.6), is given by m

i-1

S = --~ H [ n - Z i-1

SJ]g(Yi,n)[G(Yi#)]R* '

j-1

where y~,ù = (Xz,ù - bt)/0-, xi,~ = lnz~,~,E(Y~,~) =/~»i = 1 , . . ,m with 9(') given by (2.7) and G(y) = exp{-eY}. Note that in obtaining the above approximate variance-covariance factors, we considered a first-order Taylor approximation by expanding g'(y~,~)/g(y~,~) and 9(y~,ù)/G(yz,ù) around the actual mean #z of the standardized order statistic Y~,ù.Thus the standardized variate - k « - (~ - k0-) (0-/V/~) [711 (n) -}-/¢2722(n ) -- 2kT12(n)]

is asymptotically distributed as N(0, 1). For given two points (p~, 1 - c) and (p~,fi), and degrees of censoring ql,. • .~ qm, using approximate OC curve, we find that the acceptance constant k and the sample size n are, respectively, given by k = yp«Zl fi - ypvZ~ z« -- Z l - f l

(2.8)

and the sample size n is to be obtained iteratively by

[z.

~-Zl

fi

1'

n = Lyp~ -Yp~J

['~ll(rt) -}-k2~/22(n) -- 2kT~2(n)l ,

(2.9)

where, for 0 < ~ < 1, y~ - in I- ln(1 - ~)] and z~ - q~-I (~) with ~b(.) denoting the standard normal distribution function. A simulation study carried out by Balasooriya et al. (2000) showed that the proposed procedure provides reasonably reliable results for practical purposes. Note that the total degree of censoring and the choice of progressive censoring pattern are important factors when using these large sample results. For the case of Type II censoring, in fact, one can obtain the approximate Fisher information matrix of the maximum likelihood estimators fi and ä without using Taylor approximation to expand g'(Yi,n)/9(Yi,n) and g(Yi,n)/G(yi,n) around

U. Balasooriya

464

the mean/~i. Let p[= (n - m)/n] denote the degree of censoring at the right. In this case, for obtaining F(n) in Section (2.2), one could use the variance-covariance factors given below (see Harter, 1970; Schneider, 1989) that do not require the computation of moments of order statistics: Jll(n) = 1 -p,

J12(n) = J21 (n) = F'(2; - lnp) - p l n p . l n ( - lnp), J22(n) = - ( 1 - p) -

2[r'(1; -

lnp)] + F"(2; - lnp) +

2r'(2;

- lnp)

- p l n p . l n ( - lnp)[2 + l n ( - lnp)] , where U(1;-lnp)

__f

lnp

ln(t) exp(-t)dt,

--J0

U(2; - lnp) - f - l n p t ln(t) exp(-t)dt, --J0

and F"(2; - lnp) =

f

- lnp

t ln(t) 2 e x p ( - t ) d t .

d0

In the computation of the above integrals, one can use the appropriate subroutines of NAG, IMSL or the program given by Escobar and Meeker (1986).

2.3. Step-by-step procedure for determining (n, k) The main steps involved for obtaining variables-sampling plans under progressive censoring, are given below.

Step 1. Choose two points (p~, 1 - ~) and (p~,/~) on the OC curve and determine the total degree of censoring. Step 2. Choose an appropriate progressive censoring configuration, and determine the degrees of progressive censoring ql,. • •, qm. Step 3a. For the exponential case, compute the acceptability constant k and the sample size n using the appropriate formulae given in Section 2.1 or by solving the two equations iteratively. Step 3b. For the Weibull (extreme-value) case, compute k and n using the formulae given in Section 2.2. Note that n is to be obtained iteratively and in this regard, a computer program is required to compute the moments of the progressively censored order statistics. Step 4. Round oft the value of n to the nearest integer and the desired sampling plan (n, k) is obtained. 2.4. Remark It is clear from the above discussion that orte could obtain variables-sampling plans along the same lines for any location-scale distribution. Although we have

Progressively censored variables-sampling plans for life testing

465

used the M L E s one could use, for example, best linear unbiased estimators (BLUEs) or any other suitable estimators of/~ and «. Note that from the generalized least-squares theory, the BLUEs of/~ and a of (2.1) are given by

B = _ I«TFx and 1 = Ä 1TFX , where X = 0(1,n,X2,n,... , X.m,ù)T, Y = ( X - #l)/a, « = E(Y), 12 = Var(Y), 1 is an m × 1 column vector of l's, F is a skew-symmetric matrix defined as F=E I ( I « T - - « I T ) ~ ] -I and A = (ŒT~-I«)( 1T~2-1 1 ) - (ŒT~2 11)2 . Further, Var(fi) ~ o-2(«T~-lŒ)/A,

Var(8) ~ o-2(1TE-11)/A

and

Cov(]2, ~) ~ --O'2(«Tz ll)/A (c.f., Balakrishnan and Rao, 1997). The details on the use of BLUEs in obtaining variables-sampling plans for the lognormal (normal) distribution are available in Balasooriya and Balakrishnan (2000).

3. Illustrations EXAMPLE 1. Herd (1956) discussed the estimation of mean life of gyroscopes using progressively censored data. Suppose we are interested in determining reliability sampling plans for life testing of gyroscopes. Assume that lifelengths follow the two-parameter exponential distribution and the plan requires that there is at least a 1 - c~= 95% (1 - / 3 = 90%) chance of accepting (rejecting) a batch of gyroscopes when p~ = 0.0209 (p~ = 0.0742) where p denotes the proportion of nonconforming gyroscopes in the batch. Note that here we have chosen a (p~, p~) value from M I L - S T D - 1 0 5 D (US Department of Defense, 1963). Under the given specifications, we find k = 0.0217 and m = 42. Thus, if 60% of total censoring is required, then n = m/0.60 = (2.5)m = 105. EXAMPLE 2. In an example considered by M a n n and Fertig (1973), referred to data taken during the early development of F-100 fighter aircraft, 13 aircraft components were placed on test under Type II censoring with the test terminating at the time of the tenth failure. It was fairly established that the failure data follow a two-parameter Weibull distribution. For illustration, suppose that a progressively censored sampling plan is desired given p~ = 0.001, 1 - c~ = 0.99, p~ = 0.10,/3 = 0.01, ql = 0.3 and qm = 0.2. Using

466

U. Balasooriya

f o r m u l a e (2.8) a n d (2.9), w e o b t a i n k = 4 . 5 7 8 8 as t h e a c c e p t a n c e c o n s t a n t a n d n = 25 as t h e s a m p l e size. T h u s , t o i m p l e m e n t t h e s a m p l i n g p l a n , w e t a k e a r a n d o m s a m p l e o f 25 a i r c r a f t c o m p o n e n t s a n d r e m o v e a t o t a l o f 13 = [25 x 0.5] components

s u c h t h a t R1 = 8,R2 = R3 . . . .

R l l = 0, a n d R12 = 5.

References Balakrishnan, N. and C. R. Rao (1997). Large-sample approximations to the best linear unbiased estimation and best linear unbiased prediction based on progressively censored samples and some applications. In Advances in Statistics Decision Theory and Applications, pp. 431~444 (Eds. S. Panchapakesan and N. Balakrishnan). Birkhäuser, Boston. Balasooriya, U. (1995). Failure-censored reliability sampling plans for the exponential distribution. J. Stat. Comput. Simul. 52(3), 337 349. Balasooriya, U. and N. Balakrishnan (2000). Reliability sampling plans for the lognormal distribution based on progressively censored samples. IEEE Trans. Reliab. 49(2), 199-203. Balasooriya, U. and S. L. C. Saw (1998). Reliability sampling plans for the two-parameter exponential distribution under progressive censoring. J. Appl. Star. 25 (5), 707 714. Balasooriya, U., Saw, S. L. C. and V. G. Gadag (2000). Progressively censored reliability sampling plans for the weibull distribution. Technometrics 42(2), 160 167. Cohen, A. C. (1963). Progressively censored samples in life testing. Technometrics 5(4), 327-339. Das, N. G. and S. K. Mitra (1964). The effect of non-normality on sampling inspection. Sankhya 26, 169 176. Escobar, L. A. and W. Q. Meeker (1986). Elements of the information matrix for the smallest extreme value distribution and censored data: Algorithm as 218. Appl. Star. 35, 80 86. Fertig, K.W. and N . R . Mann (1980). Life-test sampling plans for two-parameter Weibull populations. Technometrics 22(2), 165 177. Guenther, W. C., S. A. Patil and V. R. R. Uppuluri (1976). One-sided/3-content tolerance factors for the two parameter exponential distribution. Technometrics 18(3), 333 340. Halperin, M., M. I. Hamdy and P. F. Thall (1989). Distribution-free confidence intervals for a parameter of Wilcoxon-Mann Whitney type or ordered categories and progressive censoring. Biometrics 45, 509 521. Hamilton, D.C. and M.S. Lesperance (1995). A comparison of methods for univariate and multivariate acceptance sampling by variables. Technometrics 37(3), 329-339. Harter, H. L. (1970). Order Statistics and Their Use in Testing and Estimation, vol. 2. US Government Printing Office, Wahington, DC. Herd, G. R. (1956). Estimation of the parameters of a population j?om a multi-censored sample. Ph.D. Thesis, Iowa State College. Johnson, L. G. (1964). Theory and Technique of Variation Research. Amsterdam, Elsevier. Kocherlakota, S. and N. Balakrishnan (1984). Two-sided acceptance sampling plans based on MML estimators. Commun. S t a t . - Theory Meth. 13, 3123-3131. Kocherlakota, S. and N. Balakrishnan (t986). One- and two-sided sampling plans based on the exponential distribution. Naval Res. Logist. Quart. 33, 513-522. Lieberman, G. J. and G. J. Resnikoff (1955). Sampling plans for inspection by variables. J. Am. Stat. Assoc. 50, 457 516. Mann, N. R. (1969). Exact three-order-statistics confidence bounds on reliable life for a Weibull mode1 with progressive censoring. Technometrics 64, 306-315. Mann, N. R. (1971). Best linear invariant estimation for Weibull parameters under progressive censoring. Technometrics 13(3), 521 533. Mann, N. R. and K. W. Fertig (1973). Tables for obtaining confidence bounds and tolerance bounds based on best linear invariant estimates of parameters of the extreme-value distribution. Technometrics 15(2), 87 102.

Progressively censored variables-sampling plans jor liJè testing

467

Montanari, G. C. and M. Cacciari (1988). Progressively-censored aging tests on XLPE-insulated cable models. IEEE Trans. Reliab. 23, 365-372. Newcombe, P.A. and O. B. Allen (1988). A three-class procedure for acceptance sampling by variables. Technometrics 30, 415421. Owen, D. B. (1964). Control of percentages in both tails of the normal distribution. Technometrics 6, 377-387. Owen, D. B. (1969). Summary of recent work on variables acceptance sampling with emphasis on non-normality. Technometrics 11, 631-637. Rao, J. N. K., K. Subrahmaniam and D. B. Owen (1972). Effect of non-noramlity on tolerance limits which control percentages in both tails of normal distribution. Technometrics 14, 571 575. Schneider, H. (1989). Failure-censored variables-sampling plans for lognormal and Weibull distributions. Technometrics 31(2), 199-206. Schneider, H. and P. T. Wilrich (1981). The roubstness of sampling plans for inspection by variables. In Computational Statistics, pp. 281-295 (Eds. H. Büning and P. Naeve) Walter de Gruyter, Berlin, New York. US Department of Defense (1963). Sampling Procedures and Taóles for Inspection by Attributes: MIL-STD-IO5D. US Government Printing Office, Washington, DC. Viveros, R. and N. Balakrishnan (1994). Interval estimation of parameters of life/'rom progressively censored data. Technometrics 36(1), 84-91.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 Elsevier Science B.V. Ali rights reserved.

1

KY

Graphical Techniques for Analysis of Data From Repairable Systems

P e r A n d e r s A k e r s t e n , B e n g t KlefsjÖ a n d B o B e r g m a n

The TTT-plot (TTT = Total Time on Test) and its theoretical counterpart, the scaled TTT-transform, are well known and useful tools in the reliability work. These concepts were first introduced by Barlow and C a m p o in 1975. Since then several applications based on the TTT-plotting technique and intended for data from non-repairable units have been presented. In this paper we discuss how the TTT-plot and a similar plot can be used when analyzing data from a repairable system. We discuss two different graphical approaches useful to see if a power-law process is a suitable model for the data.

Notation

R(t) p(t) z(t) xj x0? wj vj

survival function = 1 - F(t) intensity function of the failure process failure rate of a non-repairable unit, i.e. f(t)/R(t) time after repair number (j - 1) to failure number j, jth inter-event time; local time jth ordered inter-event time, i.e. x(1) _< x(2) _< .-. _< x(n) time to the jth failure, i.e. Xl + x2 + ... + xj; global time

tj/tn --ln(tn-j/tn), j = 1,2, . . , n - - 1 total time on test at the jth failure time

Sj/Sù;j = O, 1, ..,n

(i/n, uj) plotting positions to get a TTT-plot

~0(u)

the scaled TTT-transform of a life distribution F(t)

1. Introduction

The TTT-plot (TTT = Total Time on Test) is an empirical and scale independent plot based on failure data. It was introduced by Barlow and C a m p o in 1975, 469

470

P. A. Akersten, B. Klefsjö and B. Bergman

together with the corresponding asymptotic curve, called the scaled TTT-transform. Primarily these tools were developed for model identification purposes. Later on the tools have proven to be very nseful in several applications within reliability. Some examples of practical applications are analysis of aging properties, maintenance optimization and burn-in optimization. The TTT-plot and the scaled TTT-transform have also been used in the design of test statistics for particular purposes and for the study of their properties. In this paper, the usefulness and clarity of the TTT-plotting technique and some related graphical techniques are studied. Applications as simple and clear graphical tools for analysis of failure time data from repairable systems are described. The graphical methods described are primarily used for the illustration of goodness-of-fit of the power-law process, one of the most used models for reliability growth of complex systems and for reliability of repairable systems.

2. Some concepts for repairable systems Suppose that we are studying a repairable system. Let xj denote the time after repair n u m b e r j - 1 to failure n u m b e r j and let tj denote the time to thejth failure, i.e. tj = x t + x2 + • • • + xj; see Figure 1. In the study of repairable systems, one characteristic of great interest is the intensity p ( t ) of the failure process. The intuitive interpretation of p ( t ) is as a measure of the probability that a failure will occur during the next At units of time in the sense that P{a failure occurs in the time interval It, t + A t ) } ~ p ( t ) A t

.

Accordingly, in the case of an increasing intensity, the successive times between the failure events tend to become shorter and shorter. In the case of decreasing intensity they will tend to successively increase. We want to emphasize here that the intensity of the failure process for a repairable unit should not be confused with the failure rate z ( t ) for a nonLoeal time times)

xl

x2

tl

x3

t2

t3

Global time (Event epochs)

Fig. 1. Some time concepts for data from a repairable system.

Graphical techniquesfor analysis of data from repairable systems

471

repairable unit. The failure rate z(t) is a conditional measure that a unit, which has survived up to t will fail during the next A t units of time in the sense that P{the failure of the unit occurs in the time interval It, t + At)]survival up to time t} ~ z(t)At . For a thorough discussion of these concepts, see Ascher and Feingold (1984). The intensity of the failure process depends on the past history of the system. This history incorporates information about failure event epochs, measured by the global time t, and times between failures, measured by the local time x, the time elapsed since the previous repair. If the intensity of the failure process is only a function of the local time x, we have a renewal process. This means that the times between failures are independent and with the same life distribution. This situation occurs if the unit is repaired to an "as-good-as-new"-state after each failure or, equivalently, replaced at failure with a new, identical unit. This model is in most situations in practice not a realistic one for repairable units. If you repair your old car after a minor failure it is certainly not in "as-good-as-new"-state. I f the intensity is constant, say equal to 2, the failures occur according to a renewal process in which the times between the successive failures are exponentially distributed with the same parameter 2. This means that the failures occur according to a homogeneous Poisson process with intensity 2. When the intensity of the failure process is a function only of the running time t, the failures may occur according to a non-homogeneous Poisson process. This situation is obtained if, for instance, we use a minimal repair policy, which means that upon failure the system is restored to the state it had just before the failure occurred. The concept of minimal repair is described for instance in Barlow and Hunter (1960). For an excellent general discussion on analysis of failure data from repairable systems and the abuse of the renewal process assumption, see Ascher and Feingold (1984). One of the most useful non-homogeneous Poisson process models is the one where the intensity function p(t) is a power of t, i.e. it can be written in the form

~/~/ ~(0) ~-~

~~~/

with parameters 0 > 0 and fl > 0. This model covers monotone increasing intensities (fi > 1) as well as monotone decreasing intensities (fi < 1). When B = 1 we get the homogeneous Poisson process. This model has been discussed by several authors, see e.g. Crow (1974), Moller (1976), Bain and Engelhardt (1986), Bain (1991), Rigdon (1989), Rigdon and Basu (1989, 1990), Park and Seoh (1994) and Crétois and Gaudoin (1998). M a n y authors include "Weibull" in the name of this process due to the fact that the intensity function in (2.1) has the same form as the failure rate z(t) of a Weibull life distribution. However, including "Weibull" in the name causes a lot of confusion since it gives an incorrect impression that the times between failures


472

follow a W e i b u l l d i s t r i b u t i o n (for a discussion, see e.g. Ascher, 1981). W e prefer to call this process the power-law process.

3. The T T T - p l o t I f we have a c o m p l e t e o r d e r e d sample 0 = x(0) _< x(~) < --- < x(n) o f times to failure f r o m n identical a n d i n d e p e n d e n t n o n - r e p a i r a b l e units, the TTT-plot o f these o b s e r v a t i o n s is o b t a i n e d in the following way: 1. C a l c u l a t e the T T T - v a l u e s

Sj = nx(1) + (n - 1)(x(2) - x(~)) + . . . + (n - j +

1)(x(j) - x(j_l) )

(3.1)

for j = 1 , 2 , . . , n (for convenience we set So = 0). 2. N o r m a l i z e these T T T - v a l u e s b y calculating

uj = Sj/S,,

for j = 0, 1 , . . , n

.

(3.2)

3. P l o t (j/n, uj) for j = 0, 1 , . . , n. 4. J o i n the p l o t t e d p o i n t s by line segments. The a c r o n y m " T T T " in the n a m e " T T T - p l o t " m e a n s " T o t a l Time on Test". The r e a s o n for this n a m e is t h a t if all the units are p u t into test at the same time, then Sj is the t o t a l test time for all the units at time x(j); see F i g u r e 2. W h e n the s a m p l e size n increases to infinity the T T T - p l o t converges ( u n i f o r m l y a n d with p r o b a b i l i t y one; see L a n g b e r g et al., 1980) to a curve n a m e d the scaled TTT-transform o f the life d i s t r i b u t i o n F(t) f r o m which the s a m p l e has come. M a t h e m a t i c a l l y the scaled T T T - t r a n s f o r m is defined as

1 2 3 j-1

1

J i"1-1

ù. x

1

J

I

1

l

I ...

X(n)

Fig. 2. The reason for the acronym "TTT" is that if all the units are put into test at the same time, then Sj is the total test time for all the units at time xo>

Graphical techniques for analysis of data ßom repairable systems

z

1-[F-~(t) R(u)du

473

(3.3)

where R(t) = 1 - F(t) is the survival function and/~ is the mean. In this paper we only use the simply proved fact that the scaled TTT-transform of any exponential distribution, independently of the scale parameter, coincides with the diagonal in the unit square. Different types of deviations of the TTT-plot from the diagonal means different deviations from the exponential distribution. For instance, it is well known (see Barlow and Campo, 1975) that the failure rate of F(t) is increasing (decreasing) if and only if the scaled TTT-transform is concave (convex). Scaled TTT-transforms of a few life distributions are given in Figure 3. If the TTT-plot is based on a sample from an exponential distribution then Ul,U2,..,un, has the same distribution as an ordered sample from a uniform distribution on [0, 1], i.e. F(t) = t, 0 < t < 1 (see e.g. Epstein, 1960). A consequence of this is that the TTT-plot then tends to wriggle around the diagonal of the unit square. The scaled TTT-transform and the TTT-plot were first presented by Barlow and Campo (1975). They used these concepts for model identification by comparing the TTT-plot with scaled TTT-transforms of different life distributions. Since then several other applications have appeared. Among these are the analyses of different aging properties and for optimization when studying different

|

I

~,,

i

i

|

ù11

Fig. 3. Scaled T T T - t r a n s f o r m s of various life distributions: (1) N o r m a l (# = 1, a - 3); (2) g a m m a , with shape p a r a m e t e r 2.0; (3) Exponential (4) L o g n o r m a l (/~ = 0, er - 1 for the log(r.v.)); (5) Pareto with R(t) -- (1 + 0 -2, t > 0 (from Klefsjö and K u m a r , 1992).

474

P. A. Akersten, B. Klef~jö and B. Bergman

replacement problems of non-repairable units. For more information about the TTT-tools and different applications of these tools, see e.g. Bergman and Klefsjö (1984, 1988, 1998), Klefsjö (1991), Westberg and Klefsjö (1993) and references in these papers. In this paper we shall use the TTT-plot and another similar graph for model identification purposes in connection with the power-law process in the analysis of data from a repairable system.

4. Plotting inter-event times If we study a single repairable unit up to time T the data normally provided consists of epoches of failure events, measured on an operation time scale or a calender time scale. In some cases the duration of repair is reported as well. However, in this paper repair times are excluded from the discussion. The sequence of event epochs, i.e. times to failures, q, t2,.. • (to = 0), gives rise to a sequence of inter-event times, i.e. times between successive failures, Xl = tl - to,

X2

=

t2 --

tl,...

(4.1)

A c o m m o n error in the analysis of data from repairable systems is to consider the set of inter-event times as a set of independent and identical distributed (i.i.d) random variables. Only in the case of a renewal process, stopped at a pre-determined number of renewals, this is a correct approach. This situation occurs when we repair the system up to the "as-good-as-new"-state after each failure. This type of repair is equivalent to a replacement with an identical unit. In other cases this approach may lead to misinterpretations and erroneons conclusions. As an illustration of this type of misinterpretation let us study the TTT-plot based on the following (fictive) times between failures (from Ascher, 1981): 43 15 177 65 27 51 32 The TTT-plot is given in Figure 4. As the plot wriggles around the diagonal the plot in no way indicates that a homogeneous Poisson process model is incorrect. N o w suppose that the times between failures instead occur in the following order: 177 65 51 43 32 27 15 I f we make a TTT-plot using these values, we obviously get the same plot as before since we use the same observations just in a different order. However, it is obvious that we now do not have observations from a homogeneous Poisson process, but from a process with an increasing intensity function since the times between failures are becoming shorter and shorter. Under fairly general assumptions concerning the failure intensity the set of unordered inter-event times from a non-homogeneous Poisson process has the same asymptotic distribution as a sample from a distribution having decreasing


475

o

Fig. 4. The TTT-plot based on a ficitive data set from Ascher (1981).

failure rate (DFR); see Akersten (1991). Simulation results give strong support to the use of the asymptotic result even for a very small number of observations. This means that a usual TTT-plot based on the inter-event times from a non-homogeneous Poisson process is expected to behave convexly. In other words, this means that if the inter-event plot does not show a convex tendency, a non-homogeneous Poisson process may be an unsuitable model. Another conclusion is that if the inter-event times from a non-homogeneous Poisson process are treated as independent and identically distributed, the resulting TTT-plot can erroneously be interpreted as supporting a hypothesis of decreasing failure rate.

5. Plotting event epochs Suppose we are studying a repairable system during a period of time T. If the failures occur according to a Poisson process, homogeneous or non-homogeneous, then the normalized failure times

wj=tj/T,

j=l,2,..,n

,

(5.1)

where ty is the time to failure number j, will have the same distribution as an ordered sample of size n from a distribution with cumulative distribution function F(t) = A(t)/A(T), 0 0. This result is due to Moller (1976). This means that we can use a TTT-plot based on Vl _< v2 _< ... _< vn-1 (instead of x(1) _< x(2) _< .-- _< x(n)) for checking the power-law process assumption. Note also that the vTvalues correspond in a sense to the tj-values in opposite order. I f our original data are in agreement with a power-law process, the TTTplot based on the vj-values is expected to wriggle around the diagonal since the TTT-plot is an estimate of the corresponding scaled TTT-transform, which in this situation is the diagonal of the unit square. This graphical idea was presented by Klefsjö and K u m a r (1992). In Figure 8 the TTT-plot based on the vFvalues for the data from Musa (1975) illustrates a fairly good agreement with the diagonal, i.e. a power-law process seems appropriate for modeling the original data. Note that this procedure is independent of the parameter fl since the TTT-plot and the scaled TTT-transform are tools which are independent of scale. •

•

,

'

j

0 0 Fig. 8. TTT-plot, based on the Musa data, after using the vj-transformation.


481

10. Another example As a further i l l u s t r a t i o n o f the two a p p r o a c h e s p r e s e n t e d to analyze d a t a f r o m a n o n - h o m o g e n e o u s P o i s s o n process we use a s i m u l a t e d d a t a set used in Klefsjö a n d K u m a r (1992). It is b a s e d on s i m u l a t i o n f r o m a p o w e r - l a w process o f size n = 52 with fi = 0.5 o r i g i n a t i n g f r o m C r o w a n d Basu (1988). F i g u r e 9 illustrates the usual T T T - p l o t a n d the wFplot b a s e d on the original d a t a f r o m C r o w a n d Basu (1988). T h e t e n d e n c y is the same as for the d a t a f r o m M u s a (1975) in F i g u r e 3. A triala n d - e r r o r investigation with different values o f fi gave t h a t the test statistic C was

1

1

0

Fig. 9. The TTT-plot (to the left) and the w;plot (to the right) based on a simulated data set of size 52 from a power-law process with fi - 0.5 (data from Crow and Basu, 1988).

1

0 Fig. 1Õ. The TTT-plot (to the left) and the wj-plot (to the right) of the Crow and Basu data, after using the transformation t°54.

482


minimized when fl = 0.54. The TTT-plot and the wj-plot for the data transformed b y t 054 are presented in Figure 10 and, as is expected, they are both wriggling around the diagonal. As a comparison, the TTT-plot and the wj-plot of the Crow and Basu data, using the transformation with the correct value fi = 0.5 are given below. The two TTT-plots in Figures 10 and 11 are very close to each other. Likewise the two wiplots look very much the same. Figure 12 presents the TTT-plot for the vfvalues based on the simulated power-law process data from Crow and Basu (1988). As expected, the agreement with the diagonal is good in Figure 12 as well as in Figures 10 and 11.

1T

0 0 Fig. 11. TTT-plot (to the left) and ws.-plot (to the right) of the Crow and Basu data after using the transformation t°5°.

Fig. 12. The TTT-plot based on the Crow and Basu data after using the vKtransformation.

Graphical techniques for analysis of data from repairable systems

483

11. Conclusions and comments

Klefsjö and Kumar (1992) noticed that given a set of data from a renewal process (in particular with a small coefficient of variation) it was difficult to outrule a power-law process by using a TTT-plot based on the vj-values. However, if we combine the TTT-plot based on the vj-values with the combination of inter-event times TTT-plot and event epochs wj-plot, we will have a more sensitive tool. It can be used to illustrate the adequacy of the power-law process model and show the effect of different choice of model parameters.

References Akersten, P. A. (1991). Repairable systems reliability studied by TTT-plotting techniques. Ph.D. dissertation, Division of Quality Technology, Linköping University of Technology, Sweden. Ascher, H. E. (1981). Weibull distribution versus Weibull process, in Proceedings from Annum Reliability and Maintainability Symposium, pp. 426429. IEEE, Piscataway. Ascher, H. E. and H. Feingold (1984). Repairable Systems Reliability. Marcel Dekker, New York. Bain, L. J. (1991). Statistical Analysis of Reliability and Life-testing Data, 2nd edn. Marcel Dekker, New York. Bain, L. J. and M. Engelhart (1986). On the asymptotic behaviour of the mean time between failures for repairable systems. In Reliability and Quality Control, pp. 1-7 (Ed. A. P. Basu). Elsevier, Amsterdam. Barlow, R. E., D. J. Bartholomew, J. M. Bremner and H. D. Brunk (1972). Statisticallnference Under Order Restrietions. Wiley, New York. Barlow, R. E. and R. Campo (1975). Total time on test processes and applications to failure data analysis. In Reliability and Fault Tree Analysis, pp. 451481 (Eds. R. E. Barlow, J. Fussell and N. D. Singpurwalla). SIAM, Philadelphia. Barlow, R. E. and L. Hunter (1960). Optimum preventive maintenance policies. Oper. Res. 8, 9~100. Bergman, B. and B. Klefsjö (1984). The total time on test concept and its use in reliability theory. Oper. Res. 32, 596 606. Bergman, B. and B. Klefsjö (1988). Total time on test transforms. In Eneyelopedia of Statistical Sciences, Vol. 9, pp. 297-300. Wiley, New York. Bergman, B. and B. Klefsjö (t998). Recent applications of the TTT-plotting technique. In Frontiers in Reliability, pp. 47-61 (Eds. A. P. Basu, S. K. Basu and S. Mukhopadhyay). World Scientific, Singapore. Cox, D. R. and P. A. W. Lewis (1966). The Statistieal Analysis ofSeries ofEvents. Methuen, London. Crétois, E. and O. Gaudoin (1998). New results on goodness-of-fittests for the power law process and application to software reliability. Int. J. Reliability, Quality, Safety Eng. 5, 249 267. Crow, L. H. (1974). Reliability analysis for complex, repairable systems. In Reliability and Biometry, pp. 379410 (Eds. F. Proschan and R. J. Serfling). SIAM, Philadelphia. Crow, L. H. and A. P. Basu (1988). Reliability growth estimation with missing data - II. In Proeeedings from Annual Reliability and Maintainability Symposium, pp. 248553. IEEE, Piscataway. Epstein, B. (1960). Tests for the validity of the assumption that the underlying distribution of life is exponential. Part I. Teehnometrics 2, 83-101. Feigin, P. D. (1979). On the characterization of point processes with the order statistics property. J. Appl. Prob. 16, 297 304. Hollander, M. and F. Proschan (1975). Tests for mean residual life. Biometrika 62, 585 593. Hollander, M. and F. Proschan (1980). Tests for mean residual life. Amendments and corrrections. Biometrika 67, 259.

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Klefsjö, B. (1991). TTT-plotting a tool for both theoretical and practical problems. J. Stat. Plann. Inf 29, 99 110. Klefsjö, B. and U. Kumar (1992). Goodness-of-fit tests for the power-law process based on the TTT-plot. IEEE Trans. Reliab. 41, 593-598. Langberg, N. A., R. V. Leone and F. Proschan (1980). Characterization of nonparametric classes of life distributions. Ann. Prob. 8, 1163-1170. Musa, J. D. (1975). A theory of software reliability and its applications. IEEE Trans. Software Eng. 1, 312-327. Moller, S. K. (1976). The Rasch-Weibull process. Scand. J. Stat. 3, 107-115. Park, W. J. and M. Seoh (1994). More goodness-of-fit tests for the power-law process. IEEE Trans. Reliab. 43, 275-278. Rigdon, S. E. (1989). Testing goodness-of-fit for the power law process. Comm. Star. Part A - Theory Meth. 18, 46654676. Rigdon, S. E. and A. P. Basu (1989). The power law process - a model for the reliability of repairable systems. J. Qua•ty Technol. 21, 251-260. Rigdon, S. E. and A. P. Basu (1990). The effect of assuming a homogeneous Poisson process when the true process is a power law process. 3". Quality Technol. 22, 1 t 1-117. Westberg, U. and Klefsjö, B. (1993). TTT-plotting for censored data based on the piecewise exponential estimator. Int. J. Reliab. Quality Safety Eng. 1, 1-13.


1 ~7 /

A Bayes Approach to the Problem of Making Repairs

Gary C. McDonald

The question to which this paper is addressed is simply: ifa given item fails to meet specified requirements as determined by an appropriate test, what repairs (assumed finite in number) should be made in order to insure its passing at minimum average cost? The knowledge of an a priori distribution, i.e., the frequency of need for the various repair combinations, is assumed known. Thus, Bayes decision procedures are developed which map an observation, the output of an appropriate test, into the collection of all possible repair combinations. Two numerical examples are worked in detail - one in which the observation is univariate, another in which the observation is bivariate. Another related question also discussed in this paper is: How much is the test actually worth in terms of reducing the overall risk?

1. Formulation of the problem An attempt is made here to present a meaningful and valid decision theory formulation for certain testing and subsequent repair problems. The question to which this formulation is addressed is simply: if a given item fails to meer specified requirements as determined by an appropriate test, what repairs should be made in order to insure its passing at minimum average cost (or loss)? Throughout this discussion the items under consideration are assumed homogeneous within classes (states of nature); and hence, given the state of nature, the data resulting from the tests may be regarded as independent identically distributed realizations on an underlying random variable (or possibly random vector). Our goal then is to formulate good decision procedures which map an observation into the set of all possible repair combinations. Let R designate the set of all possible repairs which may be made on the item and let the number of such repairs be k, i.e., R = { r l , . . . , r k } , where ri is t h e / t h possible repair. A typical repair ticket would then list several of these repairs which should be made, i.e., a repair ticket is one of the 2k possible subsets of R. Let S = {Sl,S2,.. ,s2k} denote the collection of all possible subsets of R, so a repair ticket is now simple an element of S. 485

G. C. McDonald

486

The state of nature will be denoted by Q = { 0 1 , . . » 02k }. The state Oi will be interpreted as the item does in fact require only those repairs which are specified by the subset si. The action space will be designated by A = {a 1 , .., a2k} and the action ai is that action which repairs the item according to the exact specifications ofsi. Besides the costs of parts, labor and retests required in making repairs, one may also incur other losses in such situations. Less tangible losses might be an inconvenience of storage required in making certain repairs, a loss of prestige among contemporaries due to making too many (or too few) repairs, etc. This composite loss of utility if 0i is the true state of nature and one takes action aj is denoted by L(Oi, aj), 1 4.0.

It is interesting to note that in this example the Bayes strategy never takes the all-inclusive action a4. The Bayes risk r*(9 ) can be obtained from Eq. (2.2); in this example, a convenient form of the risk is given by

r*(9) = Z

P(d*(x)

=

ai

and 0 = Oj)

(3.2)

.

1«_i,j«4 i¢j,4

Making use of the normality of f(x]O) and letting ~b(.) denote the cumulative distribution function of the standard normal distribution, the first term in this summation is seen to be

P(«*(x)

=

a 1 and 0 ----02) = PO( ~,-" . . . . . . .

a

-,~

3 .....

r

Fig. 1. Graphical display of computations in Table 1 determining the Bayes decision function.

where f(x]01) = (27c)-1e -[(xz-2"l)2+(x2-22"O)2]/2, f(xl02) = (2~) le-[(xl-2"4)2+(x2 230)2]/2, f(x]03)

=

(27z)-le -[(x'-27)2+(x2-2°°)21/2,

f(xl04 ) = (27r)-le-[(x~-3.0)z+(xe-30.0)~l/2 .

(4.2)

A Bayes approach to the problem ofmaking repairs

493

34 33 32 31 30

Take Action a 4

29 28

27 26 25 X2 24

Take Action a 2

23 I

22 21 20 19

Take Action a 3

18

17 16 15

J

0

i

1

i

2

i

3

i

4

5

i

6

i

7

L

8

i

9

10

i

11

[

12

i

13

14

XI Fig. 2. Graphical display of Bayes decision function for the bivariate observation example.

The expressions given in (4.1) have been evaluated for various values ofxl and x2. These computations lead to the construction of Figure 2 which yields (approximately) the Bayes decision function. In Figure 2 the (xl, x2) space is divided into several regions. The lower region (denoted by RL) has the property that for each point x in this region,

~--~ L(O, a3)f(xlO)g(O ) = min ~-~ L(O, a)f(xlO)g(O ) . Q

(4.3)

G. C. McDonald

494

The middle and upper regions ( R M and Ru) have the same interpretation with a3 replaced by a2 and a4, respectively. In other words, the (approximate) Bayes decision function is given by

d* (x) =

{

a4

if x ~ upper region, Ru

a2

if x E middle region, RM

a3

if x E lower region, RL .

(4.4)

In this example the Bayes strategy never takes the "do nothing" action ai. The boundary curve between the a2 region and the a3 region, in this case, is a straight line expressed as x2 = 0. lXl q- 21.341. The boundary curve between regions a2 and a4 appears linear but actually has some mild curvature. The Bayes risk r*(9) can be obtained as in the previous example, i.e.,

r*(g) = ~

L(Oj,ai)P(d*(x)

=

ai

and 0 = Oj)

.

(4.5)

l_ 1, the same condition needed for propriety of 7c(#jtrest ). The implementation of the general methodology presented in Section 2.3 is in no way restricted to formulations which yield tractable results. The examples chosen here serve merely as illustrations. The strength of the simulation approach can be efficiently exploited to expand the application regime to include, for example, mixture densities. This allows a more flexible choice of distributions for component lifetimes. For instance, one may choose the base distribution fj0 for the (possibly transformed) jth component lifetime to be t with vj degrees of freedom conforming to J)(xj) (x [1 + vTl'cj(x j - [2j)21-(vy+l)2. Although t-density is not log-concave, it is a scale mixture of normal densities which are log-concave. The M C M C approach invokes a new step involving generation of additional latent variables in order to handle such mixture structure. We note here that the proposed method of analysis results in the generation of a large number of latent variables and one should cautiously monitor the convergence of the resulting sampler. 2.3.2. Interval censoring Interval censoring occurs when time of failure is only known to fall within an interval. This arises quite naturally in longitudinal clinical trials, where subjects are monitored at periodic intervals. In industrial experiments, the premise is not only practical, but often time it is also the only feasible sampling scheine available to the experimenter. While multiple inspection at prespecified times per item is quite common, there are various examples of quantal response study in reliability literature, where each unit is inspected only once and its status (failed or running) recorded. Once again, we consider a life-testing experiment with n units and K competing risks acting on each unit. The observations are divided into three groups: (i) observation i ~ D if its time to failure is observed to be ti with cause of failure narrowed down to MRS Si, (ii) i E C if time to failure is right censored at time ti; and (iii) i E E if time to failure ~ is interval-censored in the interval [ai, bil with cause of failure narrowed down to MRS &. Of course, interval censoring is the most general kind of censoring in the sense that it includes the cases of (a) right

A. Sen, S. Basu and M. Banerjee

534

censoring (bi = oo); (b) left censoring (ai = --oo); (C) exact failure times known (ai = bi). We, however, separate the groups in our treatment since it helps understanding the distinct roles of each of D, C, and E. For the first two groups, i E D and i E C, the likelihood contributions, are, respectively, Li = ~j~si{fj(ti) I~e~j~(ti)} and Li = [I~=1 ~(ti), as before. On the other hand, we have, for i c E, Li

Z jcsi

I,

J)(t) H Æ l ( t ) dt . ~¢j )

(2.17)

Note that, with the location-scale structure discussed in this section, B(t) = zjfj° ( z j ( t - @)). The evaluation of Li, in turn, involves evaluating the integral in (2.17) which is, in general, analytically intractable. Basu et al. (2000) propose a simulation-based approach that exploits the structure in (2.17) and present a viable alternative to direct evaluation of Li in the general location-scale framework.

3. Non-parametric methodologies In contrast to the parametric models and methods investigated for analyzing masked system failure data, the non-parametric approaches have been explored to a much lesser extent, especially for reliability applications. Much of the initial investigation in the non-parametric framework concern bioassays for animal carcinogenicity. Masking often arises due to the disagreement among veterinary pathologists as to the reliability of cause of death information. In the basic framework, each animal's life history is modeled as a four-state stochastic process with an initial alive and tumor-free state, a transient alive and tumor-bearing state and two absorbing states (death with or without tumor). Dinse (1982), Kodell and Chen (1987) focus on non-parametric estimation of survival functions associated with the cause (tumor) of interest as well as the other cause(s) lumped in a single group. Racine-Poon and Hoel (1984) formally account for the uncertainty in diagnosing the exact cause of death by assigning a score on the diagnostic probability beilag correct. The score could be arrived at by a consensus among the pathologists or from past data. Under this setup, Racine-Poon and Hoel (1984) develop generalized Kaplan Meier estimators for the cause-specific and overall survival functions. Dinse (1986) obtains non-parametric MLEs of various quantities such as cause-specific hazard rates, disease prevalence rates, and expected proportion of deaths due to the disease among the entire collection of deaths where the disease was present. Dinse's (1986) treatment, however, involves the situation where deaths may occur due to a cause other than the disease of interest, even when the disease is present; a framework harder to conceive in the reliability framework. Although most of the development is illustrated for two competing risks, where one risk represents the key cause and the other combines everything else, Dinse (1986) indicates a generalization to the multi-component case and describes how EM algorithm can be used to generate the MLEs of the

Analysis of maskedfailure data under competing risks

535

quantities of interest in this case. Under the same setup, Dinse (1988) further presents non-parametric estimation of tumor incidence rate which corresponds to the rate of tumor onset among live tumor-free animals. Recently, Ebrahimi (1996) has noted the effect of misdiagnosing the cause of death under a competing-risks framework. In the reliability framework discussed in this article, Schäbe (1994) exploits the strong theoretical foundation of counting process methodologies to derive nonparametric estimators ofFj based on masked system life data. Let pij(t) denote the probability that at time t, failure of component i will lead immediately to a succeeding failure of component j. It is assumed that Pij(t) = Pji(t) = Pij ,

an assumption needed to prevent non-identifiability and is implicit in the symmetry condition (C2) stated in Section 2. Let Hi, as before, denote the cumulative hazard function for t h e / t h component. Weakly consistent estimators ofp~j and H/ are provided by

~,s

-

,q,(t)

~~s

nij + ni _}_ nj

= H * x("1

'

~ , x ~:t,«< v'"~°~' - , 2d/(tk) - Pij) --'t-7~~'~ ,

(3.1)

(3.2)

jTAi

where di(t») = number of units that fail due to risk i alone at time th and n . ( & ) = number of unfailed and uncensored systems just prior to tk. Here 0 < tl < t2 < . . . denote an enumeration of the distinct system failure times. Eqs. (3.1) and (3.2) are friendlier and more easily interpretable versions of Eqs. (6) and (7) of Schäbe (1994). A non-iterative estimator F/ of F/ can be constructed from the relation =

1]

( 1 - d~q~(tk)) ,

k:tk 14/)

E[Cb(L, W]XI = x)] =

{

Cb+Cbx/W+E[Cb(L-x,W)]

if 0 _ < x < W ,

2Cb + E[Cb(L - x, W)]

if W < x < L,

Cb

if X > _ L (29)

On removing the conditioning, this becomes ElCh(L, W)] =

/0 ~ Cb m i n { x / W ,

1}f(x)dx +

/0 ~E[Cb(L -

x, W)f(x)dx (30)

This can be rewritten as (for details see Blischke and Murthy, 1994) ElCh(L, W)]

Cb{1 + F ( L ) - [ F ( W ) - # w / W ]

x [1 + M ( L -

W)]

«~~,~~,~,~»/~~~/0 ~-~,«~,«,~,~,~,} ,~~, The manufacturer's life cycle, Cm(L, V/) is the cost of supplying the items over the life cycle. Since failures occur according to a renewal process, we have E[Cm(L, W)] = Cs[1 -]-M(L)] .

(32)

The manufacturer's expected profit E[P(L, W)] is the difference between expected income (which is the expected cost to the buyer) and expected cost. As a result, E[»(L, W)] = < C b ( L , W)] - G[1 + M(~)] .

(33)

Warrantyand reliability

503

EXAMPLE 5. Consider the battery of Example 4. Since the failure distribution is exponential, it is possible to obtain an analytical expression for E[Cb(L, W)] given by (35). The result is

(

E[Cb(L,W)I=Cb l +e ;~w~ ( 2 L - 1 )T-W(1-e

)~w)

Let the warranty period be W = 1. The expected life cycle costs to the buyer and the manufacturer are given in Table 3 for 2 = 0.05 and 0.4 per year. Suppose that L = 10 years, Cb $50 and Cs = $30. Then for 2 = 0.5, the expected life cycle cost to the buyer and manufacturer are $237.73 and $180.00, respectively.

3.2.3. Policy 4. We carry out the cost analysis based on the second approach discussed in 3.1.3. The usage by time t is given by (4) with R a random variable with function k(u). We assume that failures are repaired minimally. As a result, over the warranty period, conditional on R = u, occur according to homogeneous Poisson process with intensity function given by

Section density failures a non-

r(tlu) = Oo+ 01t + 02u + 03ut .

(34)

3.2.3.1. Expected cost per unit to manufacturer Note that conditional on R = u, the warranty expires at time W if the usage rate u is less than ~ = U/W and at time Tu = U/u if the usage rate is u > ~. The expected numbers of failures over the warranty period for these two cases are

EEN(W, U)]uI =

/0~r(tlu)dt

and

EIN(W, U)lu ] =

/0~~r(tlu)dt .

(35)

respectively. On removing the conditioning, we obtain the expected number of failures during warranty and, using this, we find the expected warranty cost E[Cm(W,U)] to be

E[Cm(W, U)] =er { ~~ [foWr( t]u)dtl k(u)du + .~~

Ifor°r(tlu)dtlk(u)du}. (36)

Table 3 Expected warranty costs for Example 5 2

E[Cb(L,W)]

E[Cm(L,W)]

0.5 0.4

(0.8195 + 0.3935L)Cb (0.8461 + 0.3297L)Cb

(1 + 0.5L)Cm (1 + 0.4L)Cm

D. N. P. Murthy and W. R. Blischke

564

EXAMPLE 6. We consider an auto warranty for which the unit for usage U is 104 miles and for W the unit is years. Thus W = 1 and U = 2 corresponds to a time limit of 1 year and a usage limit of 20,000 miles, and the unit for R is 104 miles per year. Let k(u) = 0.2 for 0 5. This implies that R has a uniform distribution with a mean usage rate of 2.5 (or 25,000 miles per year). We consider three cases. In the first, the failure intensity is affected by age, usage rate and the total usage. In this case all the four parameters in (32) are nonzero. For the second case, we assume that the failure intensity is not affected by the usage rate, so that 02 = 0 and only age and usage influence the failure rate. Finally, for the third case, we assume a failure intensity that is not affected by either usage rate or total usage, so that 02 = 03 = 0, and only age influences the failure rate. We assume the following parameter values: Case (i): 00 = 0.003,

01 = 0 . 0 0 7 ,

02 = 0.003,

Case (ii): 00 = 0.003,

01 = 0.007,

02 = 0.000,

Case (iii): 00 = 0.003,

01 = 0.007,

02 = 0 . 0 0 0 ,

03 = 0.003.

03 = 0.003. 03 = 0.000.

The ratio of expected warranty cost to repair cost (obtained by solving (36) numerically) is shown for each case in Table 4 for U = 1.0, 1.5, and 2.0 and W = 0.50, 1.00, 1.50, and 2.00. As can be seen, for all three cases, the expected warranty costs increase with W and/or U increasing. 3.2.3.2. Expected life cycle cost A life cycle cost analysis can be carried out in a similar manner to that for Policy 1. The results can be found in Blischke and Murthy (1994).

Table 4 Expected warranty costs for Example 6 -1, Uand W--+

0.50

1.00

1.50

2.00

0.0024 0.0031 0.0035

0.0034 0.0048 0.0061

0.0040 0.0059 0.0070

0.0046 0.0069 0.0091

0.0014 0.0017 0.0018

0.0021 0.0027 0.0033

0.0026 0.0036 0.0043

0.0031 0.0043 0.0053

0.0017 0.0021 0.0023

0.0022 0.0029 0.0034

0.0025 0.0033 0.0040

0.0031 0.0036 0.0044

(a) Case (i) 1.0 1.5 2.0

(b) Case (ii) 1.0 1.5 2.0

(c) Case (iii) 1.0 1.5 2.0

Warranty and reliability

565

3.2.4. Other policies Blischke (H, Chapter 10) deals with one-dimensional free replacement and other rebate-related warranties and Patankar and Mitra (H, Chapter 11) deal with onedimensional pro-rata policies. One-dimensional combination policies were first studied by Nguyen and Murthy (1984). For further details, see Blischke (H, Chapter 12). Most of the warranty cost analysis is based on expected costs. Thomas (1989) does a comparative evaluation of the expected warranty cost with different failure distributions and same mean time to failure. Blischke and Vij (1997) explore this topic further through numerical studies. Sahin and Polatoglu (1998) deal with the probabilistic characterization of the warränty costs for the simple free replacement policy. A two-dimensional point process formulation for the study of two-dimensional free replacement policies was first proposed by Murthy et al. (1995a, b). The onedimensional approach was developed independently by Moskowitz and Chun (1994) and Iskandar (1993). Iskandar et al. (1994) deals with two-dimensional combination policies. Many different policies with different shapes for the warranty region have been proposed and studied - see, Iskandar (1993) and Singpurwalla and Wilson (1993). See Moskowitz and Chun (1996) and Wilson and Murthy (1996) for more on the cost analysis of two-dimensional warranties. Cumulative warranties were first studied by Guin (1984). Chapter 6 of W dealswith the cost analysis of several such policies. See also Zaio and Berke (1994). RIW policies are more complex, as they involve reliability development, testing, etc., and guarantees on M T T F and other reliability-related elements. Most models that have been developed are relatively simplistic in nature, for the most part ignoring stochastic aspects. This is a result of the complexity of the situation being modeled. On the other hand, in spite of the lack of modeling results, empirical evidence has shown that RIW has often proven to be cost effective. Additional analysis in this area is needed. A review of RIW models can be found in Chapter 7 of W. 3.3. Warranty cost analysis for used products We consider the case where A, the age of item at sale, is known and all failures over the warranty period are minimally repaired.

3.3.1. Policy 9 Ler TCj denote the cost of rectifying the firstj failures subsequent to the sale. It is given b J

TCj = Z

Cri

(j = 1 , 2 , . . ) .

(37)

i--1

where Cri is the cost of t h e / t h failure rectification. Since the total cost of claims to the dealer over the warranty period is limited to CT, we have for t h e / t h failure, the cost to the dealer is

Di = min{Cri, (CT - TC(i_I)}

(38)


566

and the cost to the buyer is

Bi = max{0, (TC, - Cx)}

(39)

Let TDj denote the cost to dealer associated with the first j failures. This cost is J

IDj = Z D i

(j = 1 , 2 , . . )

.

(40)

i-1 Similarly, let TBj denote the cost to buyer associated with the firstj failures. This cost is

TBj = ~ ~ B i

(j = 1 , 2 , . . )

.

(41)

i=1 with D~ and B~ given by (38) and (39), respectively. We take TDo and TBo to be 0. The w a r r a n t y can cease either when the c o m p o n e n t reaches age (A + W), with the n u m b e r o f failures being N(W; A), or earlier at t h e j t h failure if Tl) O _ 1) < Cx and TDj > Cx a n d j < N(W; A). This is shown in Figure 2. Let Z w be the cost o f rectifying all the failures over the period [0, W) and let V(z) be the distribution function for Zw. Then OO

V(z) = Z

P{Zw Fig. 2. Limit on total cost warranty policy.

W


567

Note that Zw, conditional on N(W; A) = j, is the sum o f j independent and identically distributed random variables with distribution G(z). Since N(W, A) is Poisson distributed we have

~~z~ ~[~~~~z~/~ ~

~~,~d, exp~

~/,/d,/~~'

~4~~

where GO~(z) is the j-fold convolution of G(z) with itself. The cost to the dealer is Z w if Z w < CT and CT if Z w > CT. In the former case, the warranty ceases at W and in the later case it ceases before W. Let Cd(W;A) denote this cost. The expected value of this is given by

E[Cd(W;A)] = ùf0cT zv(z)dz + CT~-(CT) ,

(44)

where v(z) is the density function ( = dV(z)/dz) associated with the distribution function V(z). The cost to the buyer over the period [0,W) is max {0, Z w - C T } . Let [Cb(W;A)] denote this cost. The expected value of this is given by

E[Cb(W;A)I =

(z - CT)V(z)dz .

(45)

T

When g(c) is an exponential density function and r(t) = 2 (i.e., failures occur according to stationary Poisson process), v(z) can be obtained analytically. (See Cox, 1962 for details.) For general r(t), it is not possible to obtain v(z) analytically and a simulation approach taust be used. EXAMPLE 7. Let the parameter values for the intensity function (given by (3)) be /~ = 2 and 2 ~- 0.443 and the parameter for the cost distribution (given by (6)) be p = 0.01. This implies that the expected cost of each repair is /~c = $100. Let CT = $450. The expected cost to the dealer, E[C«(W; A)], and the expected cost to the buyer, E[Cb(W; A)], for various combinations of A and W are shown in Table 5. (The values were obtained by simulation. For further details, see Chattopadhyay, 1998.) The sum of these two costs (i.e., cost to the dealer and buyer) is the expected warranty cost to the dealer under Policy 1 as the total cost is borne by the dealer. This is shown in Table 6.

3.3.2. Other policies Chattopadhyay (1999) is the first to deal with the cost analysis of warranties for used items. It studies many different types policies and also examines the case where the dealer has the option to upgrade an item (through overhaul, replacement of worn out components) prior to the sale.


568

Table 5 Expected warranty costs for Example 7 (Policy 9) W

A 1

2

5

6

7

[Cd(W;W)] 62.15 80.74 1 2 8 . 9 8 162.45 199,31 239.33 261.72 304.99

97.87 192.19 273.41 341.84

116.05 220.21 306.67 368.11

132.73 247.45 333.28 389.11

(b) Expected buyer's cost E [Cb(W;A)] 0.5 0.33 0.84 1.84 3.23 1.0 1.36 4.01 8.21 14.55 1.5 3.75 11.82 23.80 41.03 2.0 9.98 26.66 5 1 . 3 1 86.14

4.88 22.30 62.00 128.44

6.67 33.25 88.80 176.75

9.16 45.33 119.53 232.15

(a) Expected dealer's cost E 0.5 23.30 43.02 1.0 55.90 94.00 1.5 96.46 150.59 2.0 146.36 210.67

3

4

Table 6 Expected warranty cost for Example 7 (Policy 1) W

0.5 1.0 1.5 2.0

A 1

2

3

23.63 57.26 100.21 156.34

43.86 63.99 98.01 137.19 1 6 2 . 4 1 223.11 237.33 313.03

4

5

6

7

83.97 177 280.36 391.13

102.75 214.49 335.41 470.28

122.72 253.46 395.47 544.86

141.89 292.78 452.81 621.26

4. Engineering and management of reliability and warranty

4.1. Warranty and reliability improvement Most products consist of several components and the reliability of the product is a function of component reliabilities. If a critical component (one whose failure results in product failure) has low reliability, then a large number of failures (and claims) under warranty can result. This leads to high warranty costs. Two approaches to reducing the warranty cost are (i) building in redundancy for critical components, thereby improving the overall reliability of the product and (ii) improving the reliability through research and development.

4.1.1. Warranty and redundancy Typically, redundancy involves replication of the critical components. This possible only for components for which incorporation of such replication permissible by the functional design of the item. Building in redundancy results greater manufacturing cost per item and this is justified only if the reduction warranty costs exceeds this increase.

is is in in


569

Various types of redundancies (hot, cold and warm standby) can be used. We discuss two of them for the case where redundancy involves a module comprising two identical components. Let F(t) denote the failure distribution of component and Fm(t) denote the failure distribution for the module. 4.1.1.1. Hot standby (Active redundancy) Here the two components in the module are in use so that module failure occurs when both of them fall. As a result, the failure time for the module is the bigger of the failure times for the two components in the module. This results in

Fm(t) = IF(t)] 2 .

(46)

4.1.1.2. Cold standby (Passive redundancy) Here the module involves a switch. When the first component of the module fails, the second is switched on. As a result, the time to failure for the module is the sum of the two component failure times. In general, the switch is imperfect. Ler q denote the probability that the switch functions properly when needed. As a result, the failure density function for the module is given by fm(t; q) = (1 - q)f(t) + q f ( t ) * f ( t ) ,

(47)

where * is the convolution operator and q is the probability that the switch functions properly when needed. (q = 1 corresponds to a perfect switch.) Redundancy in the context of warranty was first discussed by Murthy and Hussain (1994). Hussain (1997) deals with redundancy and warranty and examines the optimal redundancy decisions for hot, cold and warm standbys for both FRWs and PRWs.

4.1.2. Reliability growth Reliability growth has received a great deal of attention in the reliability literature. Nguyen (1984) was the first to study reliability development in the context of product warranty. Murthy and Nguyen (1988) deal with a model to determine the optimal development based on the reliability growth model proposed by Crow (1974). In real life, the outcome (reliability achieved at the end of the development period) is uncertain. Hussain (1997) developed more complex stochastic models for reliability growth that take this into account and derives optimal development strategy that achieves a trade-off between the development cost and the reduction in the expected warranty servicing cost. 4.2. Warranty and quality control 4.2.1. Basic concepts Because of variability in manufacturing, some of the items produced do not conform to design specifications. Items that conform to the design specifications are called "conforming" (of non-defective) and those that do not are called "non-

570


conforming" (of defective) items. Non-conforming items are less reliable, and hence result in higher warranty costs. Here we look at modeling of these two types of items. Let F(t) and H ( t ) denote the failure distribution functions for conforming and non-conforming items, respectively. The failure rate for a non-conforming item is assumed to be higher than that for a conforming item over the interval [0, ec). This implies that H(t) > F(t) for all t. Modeling of the occurrence of non-conforming items depends on the type of manufacturing process used. The process to be used depends on the demand for the product and is determined by economic considerations. If the demand is high, then it is economical to use a continuous production process. If the demand is low to medium, then it is more economical to use a batch production process, where items are produced in lots (or batches). In either case, the state of the manufacturing process has a significant impact on the occurrence of non-conforming items. In the simplest characterization, the process state can be modeled as being in one of two possible states - (i) in-control and, (ii) out-of-control. When the process state is in-control, all the assignable causes are under control and, although non-conformance cannot be avoided entirely, the probability that an item produced is non-conforming is very small. The change from in control to out of control is due to one or more of the process parameters no longer being at required target values. This increases the probability that an item is non-conforming. Let Pi and Po denote the probability that an item produced is conforming when the process is in control and out of control, respectively. In general, Pi » Po. In the extreme cases, Pi--1, implying that all items produced are conforming when the state is in-control, and Po = 0, implying that all items produced are non-conforming when the process is out of control. 4.2.1.1. Continuous production Under continuous production, the manufacturing process begins in control and, after a r a n d o m length of time, it changes to out of control. When the process is in control, the probability that an item produced is conforming is Pi and that it is non-conforming is (1 -Pi). As a result, the failure distribution of an item can be modeled by a mixture of distributions, G(t) = p F ( t ) + (1 - p ) H ( t )

(48)

with p = Pi when the process is in control and p = Po when out of control. Once the process state changes from in-control to out-of-control, it remains in that state until it is brought back to in-control through some corrective action. 4.2.1.2. Batch production Here the items are produced in lots of size L. At the start of each lot production, the process state is checked to ensure that it is in control. If the process state is incontrol at the start of the production of an item, it can change to out of control


571

with probability (1 - q), or continue to be in control with probability q. Once the state changes to out-of-control, it remains there until completion of the lot. As mentioned previously, an item produced with the state in-control [out-of-control], is conforming with probability Pi [Po]. Ler No denote the number of conforming items in a lot. Note that this is a random variable. We obtain the distribution of Nc using a conditional approach. Toward this end, let N denote the number of items produced before the process changes from in-control to out-of-control. This is a random variable that can assume integer values in the interval [0, L]. The probability distribution of N is given by

P{N = n} = { qL, qn(1 - q)'

n0_-- ~~=1 Nsqt_s, do not hold for t = 1 , .., S. To avoid this, Karim et al. (200la) propose an estimator using the EM algorithm. The EM algorithm is a well-known algorithm for finding the M L E for incomplete data problems (Dempster et al., 1977). It consists of the E-step and the M-step. In this case these are: E-step: The conditional distribution of {r«j_s; 0 < s < j} given rj is a multinomial with sample size rj and probabilities Ns@-«/(Z~=0 N~qj_i),s = 0 , . . ,j, for j = 0 , . . , S. In general, the E-step finds the conditionally expected log likelihood of the complete data, given the observed data and the current fit of the parameters. For the linear exponential family, the E-step estimates the conditional expected values of the sufficient statistics of the complete data, given the observed data. For this model, at the (k + 1)th iteration, the E-step is r ~r (k)

ÆqL [r~«-'l~A - , ~ j /v«q)_~ . . (k), -

2-.~i=0lviq)

s = 0 , . . ,j .

(17)

i

M-step: The conditionally expected complete data log likelihood obtained in E-step is the same as (12) except for each rst replaced with Eq}kl[rst]r«+t]. Maximizing it with respect to qt, we have

Statistical analysis of reßabißty warranty data

0(k+l/ E,\-; e~~/{rs~lrs+t] t

=

S t ~,:0N~

t=O,...,S

.

597

(18)

2.4.2. Non-repairable products

For non-repairable products, rso,... ,rs,s s might be assumed to have a multinomial distribution with parameters Ns and q 0 , . . , qs s, for s = 0 , . . , S. qt can be interpreted as the probability that the lifetime of an individual is t. Marginal distributions from several heterogeneous multinomial distributions do not have a closed form in general, nor do the conditional distributions. Therefore if the EM algorithm is applied to estimate qt, a very computer intensive calculation will be required to obtain the conditional expectation of the complete data log-likelihood, given the observed data. For this problem, Karim et al. (200la) propose a Poisson approximation. This is highly accurate especially when many products are sold in each interval and a product is highly durable. If so, {rs«} are very small compared to {Ns}. Therefore the distribution of the r~t's can be approximated by a Poisson distribution with parameters N«qt,s = 0 , . . , S , t = 0 , . . , S - s. The likelihood functions based on the complete data and marginal data, and their estimating equations are the same as the NHPP model in Section 2.4.1. The computational complexity of an analysis which assumes multiple multinomial distributions is emphasized by Tortorella (1996) and Escobar and Meeker (1999). Escobar and Meeker (1999) also suggest Poisson approximations and Normal approximations. Karim et al. (200la) discuss the properties of the EM estimators derived in this section.

3. Estimation of the failure time distribution using follow-up information 3.1. Notation and assumptions

In the following sections, estimation of the failure time distribution is discussed. Let (X/, Y/), i = 1 , 2 , . . , N, represent independent, identically distributed pairs of random variables, where X/- is the variable of interest with pdf f ( x ) and survival function F(x), and Y~is some censoring variable with pdf 9(x) and survival function G(x). For example, 2(/might be the mileage to the first failure of product i, and Y/ might be the total mileage of product i dufing the warranty period. Also ler 0 represent a vector of unknown parameters, taking on values in the parameter space B. The observed quantities are (Zi, 6i), i = 1 , 2 , .. ,N : where Z/~_ min(X/, Y/), Bi =-I[X/< Y/], i = 1 , 2 , . . , N . Hefe/[.] means the indicator function of set [.]. That is, the random censoring model (e.g., Efron, 1967; Miller, 1981) is applied to the problem. The quantities Z/in the pairs (Zi, 6i) are unobserved for some is. Ler Di --= 1 if the/th product is followed up; otherwise, D i = 0 (i = 1 , 2 , . . . , N ) . Notice that Di is a known constant, not a random variable. In order to facilitate the description in this article, we shall make free use of terminology relevant to the automobile example which follows. Thus, we refer to

K. Suzuki, Md. R. Karim and L. Wang

598

X~ as mileage to the first failure of product i, and Yi as mileage in the warranty period of product i. Also, we define nu - ~ 6~ to be the number of automobiles that fail in the warranty period. The subscript u on nu means uncensored, and means ~ N 1 ( ~ will be used in this way throughout this article); nc = ~ ( 1 - 6~)Di to be the number of automobiles without failure in the warranty period but for which mileage was determined through follow-up. The subscript c on nc means censored; nl =- ~ ( 1 - 6i)(1 - D¢) to be the number of automobiles without failure that have not been followed up in the warranty period. The mileages for these automobiles have not been observed. The subscript 1 on nl means lost. Also we define N = n u + n c + n l to be the total number of automobiles and p* -- ( l / N ) ~ D ~ to be the percentage of automobiles followed up. Throughout Sections 3-5, we make the following assumptions: (1) X/ and Y/ (i = 1 , 2 , . . , N) are independent for all i; (2) The time scale of the rv's X and Y is assumed to be actual operating time (e.g., mileage, frequency, etc.); whereas the observational duration of the study is measured by calendar time (e.g., month, year, etc.); (3) The probability of the failure of a product depends only on its actual operating time; (4) All failures during the warranty period will be reported to the manufacturer. This is essential for obtaining the mileages of the failures. If there is no failure, the owner will not report the mileage in that period. Consequently, "no record of failure" means there has been no failure. Also, in this section we assume; (5) The percentage of follow-ups in the study, p*, is not equal to zero. Moreover, n c ¢ 0 and nu ~ O; (6) Individual automobiles to be followed up are selected randomly, and the correct mileages of followed-up automobiles are observed with probability 1, even if they have not failed.

3.2. Parametric approach for estimating the failure time distribution using follow-up information Under the assumptions in Section 3.1, the sampling distribution of the observed quantities (Zi, 6i), i = 1 , . . ,N, is given by N

~ - { f (Zi)G(Z~) } a,{g(Zi)p(Zi) } (1-a,)», {pr( 3i = O) } (~-a,)(1-D,) . i=l

Changing subscripts, the likelihood function becomes

L =

f(Zi)G

g(Zj)Æ

[Pr(X > Y)]"' ,

(19)

where Z i ( i = l , . . . , n u ) is the Zi conditioned on X / < Y / ( Ô i = l ) , and Zj (j = 1 , . . , nc) is the Zy conditioned on X~ > Y/ (c~~= 0) and Di = 1. Pr(X > Y) cannot be expressed in a simple closed form except for special cases (e.g., both X and Y are exponential; Miyagawa, 1982 investigated this special case assuming nc = 0). The distribution of the nl products that did not fail and for which no mileage frequencies are available is the same as that of2y. Ifwe try to estimate the

Statistical analysis of reliability warranty data

599

nl unobserved nonfailure mileages, we should redistribute these nl observations equally to the nc observed values of the Zj. Therefore, instead of [Pr(X > y)]nl, "c

1 n~/ùc

o(Zj)F(Zj)J

should be applied in maximizing L. Then we have pseudo-likelihood L** =

[~,,~,,~,~i,l~I~~«»~«,,, 1 l+ù,/~c

9

""

(20)

That is, every one of the observed non-failure data has an additional mass of nl/no along with its own observed mass of 1. If G(Z) and 9(Z) do not involve any parameter of interest, L** can be taken to be

L*= [Of(~.)] IO{Æ('j)}1+nl/nc ]

(21)

The proposed estimator is 0", the 0 in g2 which maximizes L*. 0* can be expected to have properties as good as those of the MLE, O, of 0 based on (19). This method is proposed by Suzuki (1985b). A similar approach is taken by Kalbfleish and Lawless (1988). They propose the pseudo-likelihood; L# =

Æ

P

.

(22)

This results from the fact that 1 + njnc converges to p*. Ler 0 # denote the value of 0 which maximizes (22). Even when the model for X is parametric and the model for Y is nonparametric, 0* and 0 # are valid. EXAMPLE 1. If X follows an exponential distribution Æ(x) = exp(-Zv), x > 0, we get L* =

2 exp(-2Zi

exp{-(1

+ù,/,c);,&}]

F r o m (8/62) log L* = 0, we obtain + (1 4- nl/nc)

)$ = nu

Zj j=l

Similarly, we obtain

}

/{ ..... i=1

for L #.

j=l

.

K. Suzuki, Md. R. Karimand L. Wang

600

EXAMPLE2. I f X follows a Weibull distribution Æ(x) = exp(-)xm), x > 0, we get L* =

2mZ~ -1 exp(-2Z, m)

exp -(1 + ni/nc)227

.

1_i:1 From (~/~m) log L* = 0 and (~/~2) log L* = 0, we have ~ u {(log Zi)Z: ~} + (1 + nl/no) ~ o { (log Zj)Z~ }

l / m + Z l o g Zi/nu = u

2=nu

2~Z;" + (1 + nl//nc)2cZ] n

/I~~ i~+ ,,+~,~~~,c~~:~l

,~~,

Here F u means ~i~1 and P c means ~ j =nol . The solution O* = (m*, 2")' of the preceding equations can be obtained by using the Newton-Raphson method. Similarly (1 + nl/nc) is replaced by p* for L #. The asymptotic properties of 0* and 0 # are as follows. For details and for the definition of J(0) and I(O,p*), refer to Suzuki (1985b). PROPERTY 1. Asymptotically, the solution 0* of (ô/~0) log L* = 0 coincides with the solution of 0* of (~/~0) log L1 = 0, where LI is the full likelihood that for each item it is known whether or not the item failed (i.e., p* = 1). Therefore, under regularity conditions (Zacks 1971, p. 194), 0* is a consistent estimator of 0. 0 # also has the above property. PROPERTY 2. Under the regularity conditions, v#N(O* - 0) ---+Nor (0, J(0) lI(O,p*)J(O)-l)

as N ---+oc .

Let Bp(O) represent the 100 × p percentile of the lifetime distribution of X, and assume that the partial derivatives (~/~O)Bp(O) exist. Then, PROPERTY 3. Bp(O*) becomes a consistent estimator of Bp(O), and V~(Bp(0*)- Bp(O)) --+ Not (0, Q(O)'J(O)-II(O, p*) × J(0)-IQ(0)) as N --+ ex} where Q(0) = (~/~ß)Bp(O) is the column vector of partial derivatives.

3.3. Non-parametric approach for estimating the failure time distribution using follow-up information In this section, we take a non-parametric approach. The generalized maximum likelihood estimator (Kiefer and Wolfowitz, 1956) of the survival function of fr(t)

Statistical analysis of reliability warranty data

601

of the random variable X is given and its statistical properties are described. We assume that X / a n d Y//are independent for all i, and that X~ is a discrete random variable taking values z 1 < ZK. Let z* denote the minimum of the uppermost support points of F(-) and G(.), and let K* be the maximum k that zk _< z*. To define an estimator of Æ(-), use the quantities (Zi*,d[, 6~), i = 1 , . . ,n* (n* ri •

Before time % items are run under standard environment, but this can be extended to any stress level. O(s)(Xi - zi) is interpreted as an equivalent lifetime under used-conditions. This model is a particular case of model (18). We have a linear time transformation function and O(s) is the acceleration factor. DeGroot and Goel (1979) then assumed that X has an exponential distribution with parameter 2 and derived the estimators of 2 and O(s). Note that their

Step-stress accelerated life test

635

approach does not allow one to assess the form of the stress function, and it just provides a point estimator. For loss functions of the form L()~, 2) = )~k2l(2 -- 2) 2 ,

(19)

where - 2 < k < 0 and - c o < l < +oc, D e G r o o t and Goel (1979) obtained Bayes estimators. They first assumed that 2 = 2o is known, worked with the parameter B = l/a, and considered a prior distribution of fi to be a gamma distribution with parameters r and b20. The posterior distribution of fi turns out to be again a gamma distribution with parameters rl and bi20, where rl = m + r and bi = b + ~i~(Y~ - ~i), with A being the set of indices corresponding to lifetimes of items that failed after being switched to the higher stress-level. For r + l > 0, the Bayes estimator/) with respect to the loss function L is B = "yk(rl -t- l) ~1 , sl Oo

where the function 7k is defined (for , / > 0) as

{ [

1 k+l+

( ~)1/2] 1-

1+~1

for-2 1 of the time OD with (high) probability fi, i.e., Pr(OD/CO < ôD < (D(~D) = fl , then the approximate sample size achieving this will be n ~ [NB(2~ + 1)/in(co)] 2 , where Nô is the standard (1 + fl)/2 fractile. Bai et al. (1989) extended this work to the case in which a prescribed censoring time is involved. In partially accelerated life testing, the cost of the experiment will depend on the stress change times zi (i = 1 , . . ,n). DeGroot and Goel (1979) have given optimal designs for the estimation of the unknown parameters choosing the n points zi such that the total risk is a minimum. 9. Conelusion

In this paper, we have presented an overview of step-stress accelerated lifetime tests. We have described the cumulative exposure model as well as other methods of deriving the lifetime distribution under step-stress model. We have then discussed inferential issues with step-stress testing and specifically described the maximum likelihood, non-parametric, and Bayesian approaches. Finally, we have given some details on the construction of optimal plans. References

Bagdonavicius, V. B. (1990). Acceleratedlife models when the stress is not constant. Kybernetika 26, 289-295. Bagdonavicius, V. B. and M. S. Nikulin (1997). Transfer functionals and semiparametricregression models. Biometrika 84, 365-378.

Step-stress accelerated life test

639

Bai, D. S., M. S. Kim and S. H. Lee (1989). Optimum simple step-stress accelerated life tests with censoring. IEEE Trans. Reliab. 38. Chernoff, H. (1962). Optimum accelerated life design for estimation. Technometrics 4, 381~408. Cox, D. R. (1972). Regression models and life-tables. J. Roy. Stat. Soc. Ser. B 34, 187-220. DeGroot, M. H. and P. K. Goel (1979). Bayesian estimation and optimal designs in partially accelerated life testing. Naval Res. Logist. Quart. 26, 223 235. Dietrich, D. L. and A. M. Mazzuchi (1996). An alternative method of analysing multi-stress multi-level life and accelerated life tests. In Proceedings of Annual Reliability and Maintainability Symposium. Gouno, E. (1999). An inference method for temperature step-stress accelerated life test. Quality and Reliab. Eng. Int. Jensen, F. (1985). Activation energies and the Arrhenius equation. Quality and Reliab. Eng. Int. 1, 13-17. Johnson, N. L., S. Kotz and N. Balakrishnan (1994). Continuous Univariate Distributions vol. 1, 2nd edn. Wiley, New York. Khamis, I. H. (1997). Optimum M-step, step-stress test with k stress variables. IEEE Trans. Reliab. 47. Khamis, I. H. and J. J. Higgins (1996). An alternative to the Weibull cumulative exposure model. Proceedings of the American Statistical Association, Section on Quality and Productivity. Khamis, I. H. and J. J. Higgins (1998). A new mode1 for step-stress testing. IEEE Trans. Reliab. 47. Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Miller, R. and W. Nelson (1983). Optimum simple step-stress plans for accelerated life testing. IEEE Trans. Reliab. 32. Nachlas, J. A. (1986). A general model for age acceleration during thermal cycling. Quality and Reliab. Eng. Int. 2, 3-6. Nelson, W. (1980). Accelerated life testing - step-stress model and data analysis. IEEE Trans. Reliab. 29, 103-108. Nelson, W. (1990). Accelerated Testing. Wiley, New York. Schmoyer, R. L. (1986). An exact distribution-free analysis for aecelerated life testing at several levels of a single stress. Technometrics 28, 165 175. Shaked, M. and N. D. Singpurwalla (1983), Inference for step-stress accelerated life tests. J. Stat. Planning and Inferenee 7, 295-306. Singpurwalla, N. D. (1995). Survival in dynamic environments. Stat. Sci. 10, 86 103. Tang, L. C., Y. S. Sun, T. N. Goh and H. L. Ong (1996). Analysis of step-stress accelerated-life test data: a new approach, IEEE Trans. Reliab. 45. Tobias, P. A. and D. Trindade (1986). Applied Reliability. Van Nostrand Reinhold, New York. Van Dorp, J. R., T. A. Mazzuchi, G. E. Fornell and L. R. Pollock (1995). A Bayes approach to step-stress accelerated life testing. IEEE Trans. Reliab. 38. Xiong, C. (1998). Inferences on a simple step-stress model with type-II censored exponential data. IEEE Trans. Reliab. 47.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 Elsevier Science B.V. All rights reserved.

~A

A..,~

Estimation of Correlation under Destructive Testing

Richard Johnson and Wenqing Lu

1. Introduction

Any reliability analysis of structures made of dimension lumber must overcome the problem that it is very difficult, if not impossible, to pair full sized specimens of dimension lumber so that their individual strengths are highly correlated. More particularly, the problem we address arises when, for instance, a roof of a house is loaded by wind or snow. Some of the structural members of the roof system will simultaneously undergo two modes of stress. The integrity of the roof system depends on both the bending and tensile strength of certain members. More generally, any reliability-based design approach to evaluating roofs, floors, walls or even complete buildings made of wood needs to use the joint distributions of two or three strength properties (see Suddarth et al., 1978). The idea of proof loading has been widely used in engineering (see Johnson, 1980). The scope of application was extended by Galligan et al. (1981) to provide a design of experiments for estimating the correlation of two strength properties, measured on an individual specimen. Ordinarily we would need to estimate the correlation by observing the two strength properties on an individual specimen. Unfortunately, when specimens are tested by loading to failure in any one strength mode the specimen is destroyed. Somehow, information on both strength properties must be obtained from a single specimen. Galligan et al. (1981) proposed the first practical experiments to obtain information about the correlation between two strength properties that follow a bivariate normal distribution. Their design consists of two steps:

Step 1. Load a specimen in strength mode 1, not to failure, but to a specified proof load. If the unit fails, record the mode 1 strength x. Otherwise remove the load and proceed to step 2. Step 2. Load the specimen to failure in strength mode 2. Record the mode 2 strength y. Data collected from these initial experiments provided the first ever estimates of correlation.

641

642

R. Johnson and W. Lu

Because of the concern that the proof load would damage survivors and thus change the joint distribution, the proof loads in the first experiments (Galligan et al., 1981) were purposely selected to be in the lower tail of the distribution. Although these designs provided the first estimates of correlation, they were not particularly efficient and needed to be improved upon. Most of the work to-date concerns estimation of the correlation coefficient in bivariate normal distributions. We first review the normal theory results and then give new multiple proof load designs in Section 3. Next we give some new results concerning the bivariate exponential and bivariate Weibull in Section 4. Finally, in Section 5, we present some results concerning a non-parametric approach. 2. Current designs for estimating correlation

The first approach to selecting a good proof load in the design was made by Evans et al. (1984). They estimated optimal proof loads for the bivariate normal model, for various values of correlation, by performing a simulation study. Their choice of the proof loads were those that yielded the smallest estimated variance for the maximum likelihood estimate/5 of correlation. In some applications, the marginal distributions are essentially known, so the case of known marginal distribution was considered as well as the five-parameter unknown case. De Amorim and Johnson (1986) took a more systematic approach that maximized the Fisher information about p. To ger the Fisher information, we first define T~=

{Xi ifXi _< Lx, Yi ifX~>Lx,

and {~ C/=

ifXi ifX~

Lx ,

where Lx is the proof load. We set 0 = (/fi, a 2, p, a 2, #2)'. Then, the log-likelihood function of one observation (Tl = t, CI = c) can be expressed as

-I[c

{1

2

= 2] ~ l n «2 -+

(t-/~2)2 ~ä~

ln[1 - O(ax)]

}

,

(1)

where ax = (2x - a 2 1 p ( t - #2))/~/1 - p2 and )ox = (Lx - #1)/«1. For the known marginal case, the Fisher information concerning p, or 133, is given by

1 Fco(z2 -

h3 = (1 - p2) 3

p2x)2h(ax)~b(ax)Ó(z2)dz2

,

Estimation of correlation under destructive testing

643

where qb(.) is the standard normal density and h(.) = ~b(.)/[1 - q~(.)] is the standard normal hazard rate. It is enough to consider standardized variables and then determine the standardized p r o o f load that maximizes the Fisher information for p. We call 2x the standard p r o o f load. The optimal p r o o f load for X is then: L x = o-1.~x @/A 1 .

(2)

Figure 1 illustrates the Fisher information as a function of standardized p r o o f load 2 -- 2» The information is a unimodal curve with the m a x i m u m at a positive B~opt for p = 0.1,0.3, 0.5 and 0.9• Also, if the 2 selected is within 0.1 unit of 2opt, there is only a small loss in the information about p. On the other hand if 2 = - 1

rho = 0.1

rho = 0 . 3

d d e'-

00

r-

ù0_ ci

.o_ d

E 0 r'-

o4

0

0 0

-1.0

0.0

1.0

-1.0

0.0

Lambda

Lambda

rho = 0 . 5

rho = 0 . 9

1.0

t,D

(5

E: 0

(:5 1(5 C'M

L{1

c5

ei

-1.0

0.0

Lambda

1.0

-1.0

0.0

1.0

Lambda

Fig. 1. Fisher i n f o r m a t i o n as a function of ), w h e n marginals are known.

644


is chosen as the proof load, the loss of information is great and it will be very difficult to estimate p. This Fisher information calculation shows why Galligan et al. (1981) had difficulty getting good estimates of correlation with the low proof loads used in the first experiments. A straightforward numerical integration essentially reproduces some entries in Table 2 of De Amorin and Johnson (1986). These new values are given in Table 1. Note that the optimal proof loads are in the upper tail of the distribution of strength mode 1. Further, as p increases, -~opt decreases from 0.608 to 0.221 and the corresponding Fisher information Iopt increases from 0.4111 to 5.1832. According to the information Iopt, a higher correlation can be estimated more efficiently than a low correlation. Table 1 Optimal proof loads and Fisher information for a single proof-load design marginal distributions known p

0.1

0.3

0.5

0.7

0.9

Iopt Bopt

0.4111 0.608

0.4668 0.577

0.6273 0.508

1.1383 0.401

5.1832 0.221

We now treat the case where all five parameters are unknown. The first-order partial derivatives of the log-likelihood (1) are: 8ll _ I[« = 1] ~1 + I[« = 2] 0"1 l2~'~h ~( a- x- P) ~ll ~=I[c=

z2 - 1 ~ 2xh(ax) 1]-~a12 + I [ c = 2 1 2 ù 2 ~ ,

el_~l = I[c = 2] (z2 - p2x)h(ax) ~D

(1 - p2) 3/2

(3) '

1

oz~ i{c-- 21(z2

2

'

p»(ax)

The Fisher information can be obtained from these first-order partial derivatives or from the second-order partial derivatives as in De Amorim and Johnson (1986). I=E

[(~ll) (~/1"~'] [ ~2[1] ~Õ \ ~ C B J = E - ~ - Õ ~ j

(4)

The results in De Amorin and Johnson (1986) were a surprise. We had not expected to see much change in the optimal proof loads when all five parameters are unknown. However, with five parameters unknown, the information is


645

especially small for small p and even experiments with large sample sizes are essentially non-informative. The optimal proof loads are summarized in Table 2, along with the associated element /33 of the inverse of the Fisher information matrix, for the proof-load design when all five parameters are unknown. That is, the entry i33 is the (3, 3)th element of inverse of the information matrix corresponding to p. For all of the cases presented, ,~opt is negative. The optimal loads are very different from these in the known marginal case. Table 2 Optimal proof loads and /-33, from the inverse of the Fisher information matrix. Single proof-load design (#1, a 2, P, a~, #2) unknown p

0.1

0.3

0.5

0.7

i33

87227.6 -1.140

936.255 -1.087

87.3909 -0.981

1 1 . 6 5 4 3 0.767947 -0.812 -0.516

Bopt

0.9

De Amorin and Johnson (1986) established the asymptotic normality of the maximum likelihood estimate ~. In particular, where the parameter space is specified as in Theorem 1 V~(~_p)

D»N(0,/33)

.

This justifies, at least for large samples, our approach of selecting the design which minimizes 133. We recalculate the Fisher information using De Amorin and Johnson's formulas but evaluate the integrals by a new adaptive integral routine due to Lau (1995, p. 299). It uses Simpson's rule with a Richardson Correction. All of our new calculations agree with De Amorin and Johnson (1986) up to 10 -4. The large /33 values, especially for small p, establish that the single proof-load design is inadequate for estimating p when the parameters of the marginal distribution are unknown. The original proof load design is asymmetric. This is necessary in some applications. For a full-sized lumber specimen, the compressive strength could only be measured on two machines in the US. However, a related strength may be obtained by cutting the specimen into small sections and loading these in compression on one of numerous smaller machines available. Here, the full sized specimen compressive strength must be the mode 1 strength. The two strength modes cannot be interchanged. When the proof loading could be done in either strength mode, De Amorin and Johnson (1984) suggested a symmetric proof load scheine that leads to much greater improvements. Half of the sample is first proof loaded in mode 1 and the other half is first proof loaded in mode 2. The Fisher information for the symmetric design is

Isp

=~EIp+

Ip~] ,

(5)

646


where I o is the Fisher information matrix for the single proof-load case and Ipr is the transpose of Ip along the second (SW-NE) diagonal. The optimal proof loads and the corresponding Fisher information, for the symmetric design, are summarized in Table 3. There is significant improvement over the basic single proof-load design. Most importantly, there is a sharp reduction in the value of i33, especially for small p's. Note, however, that the optimal proof loads are now positive, and so lie in the right-hand tail of the distribution. Table 3 OptimaI proof loads and 133, from the inverse of the Fisher information matrix. Symmetric single proof-load design (#1, al2, P, a2a, #2) unknown p

0.1

0.3

0.5

0.7

0.9

/33

2.9620 0.803

2.4874 0.729

1.7816 0.628

0.9560 0.496

0.2064 0.292

~'opt

De Amorin and Johnson (1986) also recommend a hybrid design, where some specimens are specifically loaded to failure in mode 2, for applications where the symmetric design cannot be implemented. 3. A new double proof load design

In cases where the strength modes are interchangeable so that a specimen can be proof loaded in either order, strength mode 1 or mode 2, it is possible to consider a three-stage design. Johnson and Lu (2000) propose the procedure: Design step 1. Load the unit in strength mode 1 up to an established maximum load L» If the unit fails, record its mode 1 strength x. If it does not fail, remove the load and proceed to Design Step 2. Design step 2. Load the unit in strength mode 2 up to an established maximum load Le. If the unit fails, record the mode 2 strength y. If it does not fail, remove the load and proceed to Design Step 3. Design step 3. Load the unit to failure in strength mode 1. Record the mode 1 strength x. The observations in this experiment consist of the breaking strength T~ of the specimen and the stage Ci at which the specimen breaks. The stage carries the information on the strength mode. To set notation, we define X~. ifX,. Lx and Y~_< Ly, X,. i f X g > L x a n d g > L y and Ci

{ {

1 ifX~ _ Lx and ~ < Ly, 3 if X/ > Lx and Y/> Ly .

647

Estimation o f correlation under destructive testing

Then, the log-likelihood function, 12, of one observation (T = t, C = c), but for two proof loads, can be expressed as 12(O;t,c) = - - ~ l n ( 2 ~ ) - I [ c

= 1] [~lncr2 q (t--20#1)2_1 .2

{1 2 (t -- #2) 2 - I [ c = 2] ~ l n % q 202

ln[1 - e(a,)]/

{1 2 (t -- #1 )2 - I [ c = 3] ~ l n % -~ 20 2

lnll - ~(ay)] } ,

(6)

where 2x - p t 72**2 2y

2x - L- x --

#1

,

ay -

O t-~~ "~Y - - -«l ,

and

Ly - #2 02

3.1. Fisher information for a double proof-load design

Two proof loads Lx and Ly will be selected to jointly maximize the Fisher information about p. To obtain the Fisher information matrix, we first calculate the first-order derivatives of the log-likelihood 12 with respect to 0 = (#1, ~2, p, az, #2)'. Define the standardized variables zi =.(t - #i)/«i, for i = 1,2, and recall that h(a) = (B(a)/[1 - ~(a)], is the hazard rate for a standard normal. The five first-order derivatives are 51q~12- I [ c = 1 ] ~7 + I [ c = 21 ol ~12 - - [[C

~ßa21

11z 2 - 1 q - I I C 2«2

= 21

lxf~7-p + i [ c = 31( ~

olph(aY)lx/~7-p 2 ,

2xh(ax) 2«2 V/1 _ p2

+ IEc = 3]~o ~~ ~ - 1

(7)

el2 _ I[c = 2] (z2 - p2x)h(ax) + I[c = 3] (Zl - p2y)h(ay) ~P (1 -- p2) 3/2 (1 -- p2) 3/2 ' ~12 _ I[c = 2] 1 ~ ~a 2 ~ßa22

-1

N

el2 _ Il« = 23 (z2

ph(ax)

pz2h(ax)

B

+

I[c

~ +I[« = 3]

=

3]

2yh(ay)

2«~lB77-p2'

h(ay)

648


Let I(e) = (I[c = 1], I[c = 2], I[c = 3])' be the vector of indicators and

then, we can write (7) as

~J22 = Mt(c)

(8)

00

The Fisher information matrix \~]

I = E

(9)

then has elements

Iik = E(milmklI[C = 1] + mi2mk2IIC = 2] + mi3mk3I[C = 3]) ,

(10)

where mik is the (i, k)th element of M. To simplify (10), consider the expectation of any integrable function 9('), which we obtain by conditioning on C. E(g(Zl)I[C

~-

1])=

F"

g(Zl)~(z1)dz1 ,

(11)

O(3

E(o(z~)I[c = 21) = P [ c = ~ e [ c

a]E[o(z~)lc = 2] 2] J

f I

]-oo g(z2)c)(z2)[1 - O(ax)]dz2 ,

(12)

where f(z2]C = 2) is the conditional density of Z2 given C = 2. Similarly,

E(g(Z1)I[C = 3])

g(zl)O(zl)[1 - ~b(ay)]dzl .

(13)


649

Applying relations (11)-(13) to each term in Iik in (10), we obtain Iik =

mi2mk2~)(z2)[1-- ~(ax)ldz2

milmkl~(Zl)dZl + O0

+

O0

mi3m~30(zl)[1 -- ~(ay)]dzl .

(14)

Notice that each Iik is the sum of 3 integrals. The first integral is either 0, because m i l = 0 for i > 3, or can be done analytically. The other two integrals can only be evaluated numerically. 3.2. Optimal selection o f double proof loads

We numerically evaluate the Fisher information for the double proof load design. Let us first consider the known marginal case. Figure 2 shows the contour plots of Fisher information as 21 and 22 vary over the fange between - 1 and 2.0. Again, p = 0.1,0.3,0.5 and 0.9 are used to represent the whole range of p. A unique maximum occurs at a pair of values with ~1 < ,~» Table 4 summarizes the optimal proof loads }q,opt and -~2,opt and the corresponding Fisher information for p = 0 . 1 , 0 . 3 , . . , 0 . 9 . The pair (21,opt, ,~2,opt) changes in a somewhat complex pattern as p increases. In each case, from Table 4, the optimal first proof load )~l,opt is 1ower than the respective single optimal proof load )~opt in Table 1. Actually, )~l,opt is negative when p = 0.9. The optimal second proof load 22,opt is rauch higher than the single optimal proof load. Is there rauch gain of the optimal double proof-load design over the optimal single proof-load design? Table 5 presents the increase and relative increase in the maximum Fisher information numbers Iopt obtained by employing the two-stage proof-loading scheine. When the correlation p is less than or equal to 0.5, the increase is less than 10%. By contrast, when p is greater than 0.5, the increase is greater than 10% and can be as large as 57%. This latter situation is the case most commonly encountered in the wood industry where two strength properties on the same specimen are typically highly correlated. We hext consider the case where (/11, a~, t3, «22, ~2] , 2~ are unknown. The optimal proof loads and 133 from the inverse of the matrix of Fisher information, for the double proof-load design, are summarized in Table 6. Comparing an optimal 133 in Table 6 with the corresponding information in Table 3 for the single but symmetric proof load design, we see that the double proof-load design is not as efficient as the symmetric single proof-load design, especially for the small/3's. However, when/3 increases to 0.9, the difference decreases to a moderate value. 3.3. A symmetric version o f the double proof load design

To make the double proof load design comparable to the symmetric single proof load design, we can symmetrize the double proof load design. In particular, when all five parameters are unknown, half of the specimens can be subjected to a double proof-load design starting with strength mode 1; and the other half of the

650


rho=0.3

rho=0.1 q

q

Ol

•-:

../..

0

Lt')

c5 ._1

0

¢5

°

"~.1 -1.0

0

0.0

1.0

0~.2 -1.0

2.0

i 0.2 0./1 0.0

1.0

Lambdal

Lambdal

rho=0.5

rho=0.9

2.0

q Ol

0 Ol LQ

t~'-

04 "0

E

.

c5

._1 0c5

.-I

o

O

o.

I-

0

-I .0

0.0

1.0

2.0

-1.0

Lambdal

0.0 1.0 Lambdal

2.0

Fig. 2. Fisher information as a fuuction of 21 and 22 when the marginals of the bivariate normal distribution are known.

Table 4 Optimal double proof loads and Fisher information for a double proof-load design marginal distributions known ,o

0.1

0.3

0.5

0.7

0.9

Iopt Bl,opt Æ2~opt

0.4296 0.509 1.245

0.4940 0.464 1.356

0.6837 0.339 1.531

1.3568 0.006 1.425

8.1459 -0.377 0.835

651


Table 5 Comparison of Fisher information numbers from the single and double proof-load designs marginal distributions knowna p

0.1

0.3

0.5

0.7

0.9

Increase Rel. inc.

0.0185 0.05

0.0273 0.06

0.0564 0.09

0.2185 0.19

2.9627 0.57

a Increase = Iopt,double- /opt,single; rel. inc, = increase/Iopt,single. Table 6 Optimal proof load and 133 from the inverse of the Fisher information matrix. Double proof-load design - (#1, er2, P, «22,#2) unknown p

0.1

0.3

0.5

0.7

0.9

i33 Bl,opt

11.6893 -0.934 1.198

7.4972 - 1.668 0.740

3.9009 - 1.618 0.666

1.5587 - 1.454 0.596

0.2548 - 1.076 0.448

B2,opt

specimens are subjected to a d o u b l e p r o o f - l o a d starting with strength m o d e 2. This design will generate m o r e i n f o r m a t i o n t h a n the s y m m e t r i c single p r o o f - l o a d design. The question is h o w m u c h m o r e ? T h e F i s h e r i n f o r m a t i o n o f the s y m m e t r i c d o u b l e p r o o f l o a d design follows the same relation (5) as does the single p r o o f - l o a d design. But n o w I o is the F i s h e r i n f o r m a t i o n u n d e r the d o u b l e p r o o f - l o a d i n g design. T h e n u m e r i c a l results c o n c e r n i n g the o p t i m a l ¤33, f r o m the inverse o f the F i s h e r i n f o r m a t i o n m a t r i x , are p r e s e n t e d in T a b l e 7. C o m p a r i n g a n y entry in T a b l e 7 with the c o r r e s p o n d i n g e n t r y in T a b l e 3, we see t h a t there is only a small gain in 133 when p is small. H o w e v e r , the gain in 133 can be s u b s t a n t i a l when p is large. N o t e t h a t w h e n p is 0.9, the r e d u c t i o n in 133 is 34%. W h e n strength p r o p e r t i e s have a high correlation, it is clearly w o r t h w h i l e to s y m m e t r i z e the double proof-loading procedure. Table 7 Optimal standardized double proof loads and /33 from the inverse of the Fisher information matrix. Symmetric double proof-load design - (#1, er2, P, er2, #2) unknown p

0.1

0.3

0.5

0.7

0.9

i33

2.8949 0.762 1.443

2.4023 0.672 1.545

1.6787 0.519 1.705

0.8254 0.208 1.569

o. 1372 -0.198 0.950

Bi,opt B2,opt

3.4. Asymptotic results

The b i v a r i a t e n o r m a l d i s t r i b u t i o n r e m a i n s a r e g u l a r case even u n d e r multiple p r o o f loading. T h e m a x i m u m l i k e l i h o o d estimate ~ satisfies v / n ( ~ - p) is as-

652


ymptotically normal with variance ¤33. Our approach has been to determine proof loads to minimize this asymptotic variance. We now present the asymptotic results for the maximum likelihood estimate under the double proof-loading design. THEOREM 1. Let O ---- {[/ii] _<M , M -1 1. Let Oo be the true value of parameter 0 and 0 be the maximum likeIihood estimate of 0 under double proof-load design. Under a bivariate normal modeI, 1. if Oo C O, Ô ~a's" Oo as n ----+ oo. 2. if Oo ~ interior of O, v~(Ô - Oo) D» N(0, i00_l)"

PROOF. Compare the log-likelihood function (6) under the double proof-load design and the log-likelihood function (1) under the single proof-load design. The third additional term in (6) is just the symmetric version of the second term with subscripts 1 and 2 interchanged. The log-likelihood function and its second order derivatives for the single proof-load case are bounded by some integrable functions (see De Amorim and Johnson, 1986). Because of the symmetry of the parameter space O and the terms in the log-likelihood function, the log-likelihood function and its second order derivatives for the double proof-load case are then also bounded by integrable functions. The asymptotic results follow according to the same argument as in the single proof-load case treated in De Amorim and Johnson (1986).

4. An example using the bivariate Weibull distribution Not surprisingly, the normal distribution does not adequately model many strength properties. Overall (see Johnson et al., 1999), the Weibull distribution seems to be a better model for the various strength properties of lumber. We illustrate our general approach by considering single proof-load designs for estimating p in the bivariate Weibull distribution having survival function F(x,y)=P[X>x,Y>yl=ex

p

-

(Ul)~+(

-

,

0 Li] . j=l

We write f'i for P[X > Li]. THEOREM 2.

i = 1,..,m.

Suppose Then

that ~-1 ¢ ~

and that

ni/n ~ 2i(0 < 2i < 1) .for

x/n(Ê[Y~]X > Li] - E[Y~tX > Li]) __+DN(O, a 2) , where the asymptotic variance is (see Johnson and Wu, 1991)

(20)


= Æi-l(1

-Æi-1)2E2[yB[Li 1 < X < Li] +

Bi-I (/z~i-1 -/~i)

-1- ,~i

--

2

-

-

655

V a r ( r e I [ x > L~-I]) B~i-1 (fr'i-1 -- fr, i)2 Var(Y~I[X > Li]) ,~/(L-1 - &)2

1

1 --

Æi-1

E[Y~I[Li_I < X < Li]IE[Y~I[X > Li-1]]

ù

Bi--1 (Æi--1 -- Æi) ~

1-Æi

. ~ . . IIL~ 1 < X < Li]IE[YfiI[X > Li]] ,

2 2i(/~~._~ ~ i ) J E [ Y

(21)

The estimator in this last theorem can be used to develop approximations to the mixed moments and the correlation coefficient. For a reasonable number of fixed proof loads m, we can numerically approximate the expectation m-1

E[X~Y fl] = ~ E[X~Y~I[Li < X Li] • Ê[reI[L, < x]] = n-7

(26)

This estimator ean be shown to be consistent as the proof loads become dense and the minimum ni --+ oo. The survival function F(Li, y), evaluated at x = proof load Li, can be estimated by the empirical cumulative distribution function (e.d.f.)

B,~(Li,y) = ~~ßl I[X~7 > Li, Y//j > y] F/i

(27)


656

The conditional distribution of specimens that survive a proof load is also of interest. In the context of lumber testing, the stronger specimens survive a proof load. If the proof load is low enough not to cause damage, the strength distribution of survivors would be stochastically larger than the original strength distributions for either mode 1 or mode 2 strength. There are two choices for estimating the denominator, P[X > Lil, of the conditional probability. We can use only the observations loaded to proof load Li or all of the observations loaded to at least that strength. Campbell and Foldes (1980) considered a path-dependent estimator of F'x,y(x,y) = P[X > x, Y > y]. Instead, we estimate F(y[X > x) using the empirical c.d.f, as the denominator

~E

kn~¢i,y) Fx,n,(Li)

F Li) = ~

(28)

Alternatively, we could estimate F(yl X > x) using a Kaplan-Meier type estimator

F-~KM n (ylX > Li)

kn,(Li,y)

(29)

where ~'xKM(Li) is the Kaplan-Meier estimator of Fx(Li) and F'n~(Li,y), Fx,ni(Li) are the values of the empirical c.d.f.'s^corresponding to P(Li,y), Px(Li) respectively. Notice that Pm(Li,y) and Fx,ni(Li) are estimated from the data (Zij, ~ij),J = 1 , 2 , . . , nj, generated exclusively under the p r o o f load Li. Further, we let: Nr be the number of survivors at strength Lr-1 and Dr the number of failures in the strength interval (Lr-l,Lrl for r = 1 , 2 , . . , m. Then, we have m nj

Nr = Z Z { I [ Z j «

> Lr-, 6jq = 11 + ( 1 - fij«)}

j--r q=l

m nj. : ZZ{I[Lr_, j - r q-1 m

> L r I],

nj

n't nj

j--r q=l

j=r q=l

and

k5~ (L,) = (-I :vr- Dr ~~

> ~j']}

(30)

nj

= ~~.[[Xj.q j=r q-1 m

<Xjq ~Lj]@Z[Xjq

Nr

Estimation o f correlation under destructive testing

We are interested in the asymptotic distributions

657

of the estimators

-~E

Bnr~(y[X > Li) and Fùi(ylX > Li). For simplicity, denote B ( y ) = Æ(Li,y),

for everyy_> 0, i = 1 , 2 , . . , m

.

(31)

THEOREM 3. L e t m

g/ =

Y/r~

~nr- ----+ ~~r(0

< ~~_< 1)

r=l

as minl o

m

ä

E

o

ur,~ :~ =-

.~-9_.

~ ~ I!:

.Eo li: i.7.

Il

,i i1

õ

II

ùo 0-

o

õ .--

l]i [i! li~

~

I!:

t~

%

o

~D

I!~ 00' L

$6"0

06'0

SffO

00" L

Og'O

$6'0

06"0

SgO

0~'0

,.-,

JaMod

JeMOd

~D

» co

_e

e_

o

.¢:0 i.rO

õ

õ »,

o

00" I.

$6"0

06"0 JeMOd

Sg'O

Og'O

00" I-

$6'0

06"0 Je,v, Ocl

Sg'O

Og'O

-= ,4

688

J. Stein and N. Doganaksoy

assumptions changed. In the following sections, we present other life-test planning approaches and point out consequences and effects of different parameter and failure time distribution assumptions. In Section 5.1, we present an approach to sample size determination using the asymptotic normality property often assumed for MLEs. Section 5.2 lists other life-test plans and approaches. Parameter estimation and distribution assumptions are explored in Sections 5.3 and 5.4, respectively. Bayesian approaches are touched in Section 5.5.

5.1. Asymptotic normaEty of MLEs Our initial effort to evaluate sample size requirements as a function of test duration was based on tables in Meeker and Nelson (1977) for the Weibull distribution. This approach uses an ML method to derive asymptotic (largesample) variances and covariances of the Weibull parameter estimates that can subsequently be used to derive approximate confidence limits for distribution percentiles. Such confidence limits are based on the large-sample normality of ML estimators. Both failed (uncensored) and unfailed (censored) test units are allowed at the end of the life-test in this approach. By following this approach initially, we generated a graph (see Figure 4) to summarize sample size estimates as a function of test duration by comparing differences in the 10th percentile between a standard and improved material. In this example, we assume 10% of the standard materials fails in two months and the new material is a onefold improvement over the standard material. L o w e r 2.5th P e r c e n t i l e of E m p i r i c a l S a m p l i n g Distdbution for the D i f f e r e n e e in 10th P e r c e n t i l e s

ä, o n

/._..

/."r- ~ .'~.................................................................................... t" !"t / " '

~"

/ " .I,/ i "l'/i • )/./

. .......................................................................................................

........

B iI • ùt r i .: ....... ....... --~........ ....

O

True Difference in 10th N=20 N=50 N=75 N=100 N=150

Percentiles

m._ I

I

I

I

5

10

15

20

Test Duration (in Months)

Fig. 4. Empirical lower 2.5th percentiles generated using the large-sample ML approach for differences in the 10th percentiles of the failure time distributions for the standard and new materials.

Life-test planning for preliminary screening of materials: A case study

689

The dashed lines in Figure 4 are lower 2.5th empirical percentiles of the sampling distribution of the difference in the 10th percentiles between the failure time distributions of the standard and new materials as a fnnction of sample size and experimental test duration. The interpretation of these empirical percentiles is: Assuming a true difference of A months between the 10th percentiles of the failure time distributions of the standard and new materials, 97.5% of the random samples of size n will yield differences, Ä, in the 10th percentiles larger than the lower 2.5th empirical percentile. The solid horizontal line (at t = 2 months for a ù small" improvement) is plotted to represent the "true" difference in the 10th percentiles between the failure time distributions of standard and new materials. Test durations and sample sizes corresponding to lower 2.5th percentiles greater than zero could be considered appropriate planning values to use in the experiment in order to determine if the new material warrants further consideration. In such cases, there is at least 97.5% probability that the point estimate of the 10th percentile for the new material will exceed the point estimate for the standard material. Upon observing the evolving nature of the problem definition, we decided to adopt the simulation-based approach which is more flexible, avoids the need to refer to tabled values, and can be easily adapted for specific needs of the program in the future (e.g., design of a multiple stress accelerated life test) where formulas are not readily available. We also believe the output from the simulation (i.e., the probability of concluding a difference in the 10th percentiles between the failure time distributions) is easier to interpret than the empirical lower 2.5th percentiles that were generated from the large sample M L approach. An estimate of the power to detect a difference can also be generated using the large sample M L approach, but one must assume the sampling distribution of the differences in the percentiles follows a normal distribution. For finite samples, it has been shown that the normality assumption is questionable (see, for example, Doganaksoy and Schmee, 1993; Ostrouchov and Meeker, 1988). 5.2. Other approaches to life-test planning

A conventional (i.e., confidence interval) approach to determine the sample size of a life-test for one population is to select a sample size such that a percentile of a statistical failure distribution is estimated with a certain level of precision. Formulas for calculating confidence intervals for a single population based on M L estimators can be adapted for the two sample comparison situation to determine appropriate sample sizes (see e.g., Nelson, 1982 or Lawless, 1982). The performance of this approach can be assessed via the simulation approach described here. Doganaksoy (1995) presented a method to determine duration of a life-test to compare two samples while the test is underway and there are unfailed (i.e., censored) units. However, that approach assumes the sample sizes have already been determined. A common type of life-test is a "zero-failure" demonstration plan. In this case, a reliability demonstration is deemed successful if all of the n units survive a

690


pre-specified time without any failures. Construction of such test plans is discussed in Meeker and Escobar (1998). 5.3. Parameter assumptions

The case study of this paper dealt with a life-test with time (Type I) censoring. In Type I eensoring, the life-test is stopped after a predetermined test time. With this type of censoring the sampling distributions depend on assumed parameter values of the underlying failure time distributions. One might, therefore, wish to consider different parameter values in simulations to assess sensitivity of results. From a practical perspective, Type I censoring is offen appealing since it makes planning of test resources (i.e., equipment, labor, material, and time) easier. Type II (failure) censoring is the case where a life-test is stopped after a prespecified number of failures is observed. Even though Type II censoring is seldom used in applications, it offers certain advantages. For example, if the main interest of the investigation is to estimate a low percentile (say, the 10th percentile) of the failure time distribution, there may be no further advantage to continuing the test after 15% of the units in the sample have failed (Meeker and Escobar, 1998, Chapter 10). Type II censoring also offers a salient statistical advantage for estimating failure time distributions that belong to the location-scale family (such as the smallest extreme value and the normal distributions that are associated with the Weibull and lognormal distributions respectively). This advantage arises from the existence of pivotal quantities whose sampling distributions do not depend on any unknown parameters (Bain and Engelhardt, 1991). Orte can then simulate the distribution of the pivotal quantity with any assumed location and scale parameter values. The test plans will not depend on the assumptions made about the parameter values. 5.4. Distributional assumptions

We did not know for certain whether the time to stress corrosion crack initiation is better characterized by a Weibull or lognormal distribution or another statistical distribution. The simulation approach can be adapted for other distributions as well. After the failure time data are obtained, the analyst can determine the appropriate distribution form for the subsequent analyses. See Meeker (1984) for consequences of incorrectly assuming a lognormal distribution when the underlying distribution is Weibull and vice versa. 5.5. Bayesian test planning

Limited historical data led us to develop a flexible approach that produced sample sizes under various assumptions. In this case study, we assumed a known shape parameter value for the Weibull distribution. A sample size determination method that is robust to parameter specifications as well as distribution assumptions employing Bayes estimation techniques is an area of future research.

Life-test planning for preliminary screening of materials. A case study

691

For Bayesian methods used in reliability data analysis see Martz and Waller (1982).

Acknowledgments

The authors thank Tom Angeliu for providing the background of this case study.

Appendix A: Pseudo-code for simulation algorithm assuming a Weibull failure time distribution I

Below, we denote the Weibull scale parameter for the failure time distributions of the standard and new materials by C~s and eN respectively. The Weibull shape parameter, /~, is assumed to equal 1.5 for all simulation conditions. Define Parameter Values for Standard Material (es,/3) Define Parameter Values for New Material (c~N,/3) Do the following for each sample size value (n - 10, 15, 20, 30, 40) Do the following for each of k = 2000 simulation runs Generate n Weibull random numbers from the Weibull distribution (Standard) Generate n Weibull random numbers from the Weibull distribution (New) Do the fonowing for the test duration in months, t = 9, 12, 15, 18, 21 Censor the n Weibull random numbers at month t from the Weibull distribution (Standard) Censor the n Weibnll random numbers at month t from the Weibull distribution (New) Estimate Weibull parameters(c2s,/~) from the n Weibull random numbers (Standard) Estimate Weibull parameters(~N,/~) from the n Weibull random numbers (New) Compute the 10th percentile of the fitted Weibull distribution (Standard) Compute the 10th percentile of the fitted Weibull distribution (New) Calculate the difference between these two 10th percentile estimates and store End End Do the following for each month t = 9, 12, 15, 18, 21 Compute the proportion of observations larger than 0 and store End End

Appendix B: Derivation of the scale parameter («) for a fixed shape parameter (~) of a Weibull distribution

The cdf, F(t), for the Weibull distribution is the fraction of the population failing by time, t, and is given in (3.1). For a fixed shape parameter value,/~ = 1.5, the scale parameter, c~, associated with 20% of the standard material failing by time t = 2 months is obtained by solving the following equation for c« 1 T h e s i m u l a t i o n s w e r e r u n u s i n g S-Plus, V e r s i o n 4.5 (1998).

692


0.20 = 1 - e -(2/c~)1'5

or

c~ =

2

= 5.44 .

(-- in(0.80)) 1/1"5 F r o m (3.2), a o n e f o l d i m p r o v e m e n t in the 10th p e r c e n t i l e r e q u i r e s d o u b l i n g the scale p a r a m e t e r w h i c h yields 10.88. Scale p a r a m e t e r s a s s o c i a t e d w i t h 4 0 % o f the s t a n d a r d m a t e r i a l failing in t w o m o n t h s a n d w i t h failure d i s t r i b u t i o n s o f the n e w m a t e r i a l a r e o b t a i n e d similarly.

References Bain, L. J. and M. Engelhardt (1991). Statistical Analysis of Reliability and Life-testing Models. 2nd edn. Marcel Dekker, New York. Doganaksoy, N. (1995). Determining the duration of a demonstration life-test before all units fail. IEEE Trans Reliab. 44, 26-30. Doganaksoy, N. and J. Schmee (1993). Comparisons of approximate confidence intervals for distributions used in life-data analysis. Technometrics 35, 175-184. Hahn, G. and W. Q. Meeker (1991). Statistical Intervals: A Guidefor Practitioners. Wiley, New York. Lawless, J. F. (1982). Statistical Models and Methods for Life Time Data. Wiley, New York. Martz, H. F. and R. A. Waller (1982). Bayesian Reliability Analysis. Wiley, New York. Meeker, W. Q. (1984). A comparison of accelerated life test plans for Weibull and lognormal distributions and Type I censored data. Teehnometrics 26, 157-171. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methods for Reliability Data. Wiley, New York. Meeker, W. Q. and W. Nelson (1977). Weibull variances and confidence limits by maximum likelihood for singly censored data. Technometries 19, 473~t76. Minitab (1997). Minitab User's Guide 2." Data Analysis and Quality Tools, Release 12. Minitab, State College, PA. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nelson, W. (1990). Accelerated Testing: Statistieal Models, Test Plans, and Data Analyses. Wiley, New York. Ostrouchov, G. and W. Q. Meeker (1998). Accuracy of approximate confidence bounds computed from interval censored Weibull or log-normal data. J. Stat. Comput. Simul. 29, 43-76. S-Plus Statistical Sciences (1998). S-Plus User's Manual, Version 4.5. Statistical Sciences, Seattle. SAS Institute (1995). JMP User's Guide, Version 3.1. SAS Institute, Cary, NC. SAS Institute (1997). SAS/QC Software." Changes and Enhaneements for Release 6.12. SAS Institute, Cary, NC. Tobias, P. A. and D. C. Trindade (1995). Applied Reliability, 2nd edn. Chapman & Hall/CRC, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook ofStatistics, Vol. 20 © 2001 ElsevierScienceB.V. All rights reserved.

'~P7

Analysis of Reliability Data from In-House Audit Laboratory Testing

Rekha Agrawal and Necip Doganaksoy

In manufacturing situations, it is often critical to be able to detect early warnings of potential reliability and field failure issues after product release. These field issues may be ones that occur later in the life of the product and thus are often quite difficult to predict during product development, and even in the early stages of manufacturing. In this chapter, we will use an estimate of the mean cumulative function (MCF) to analyze audit laboratory test data on toasters. We will show how the M C F analysis can be used as an analysis technique to monitor in-house testing of manufactured units, providing the required early warning. We will conclude the chapter with some general observations on issues encountered in applying such a technique to situations in industry.

1. Introduction

Companies all over the world are finding that in today's business environment, they must focus on delighting their customers to stay competitive. One of the biggest contributors to customer satisfaction is product reliability. Thus, a key for many companies is assuring reliability of their offerings. Improving the reliability of products is a multi-faceted process applied at all the different periods in the product life cycle. Modern reliability programs require quantitative methods for predicting and assessing product reliability, and for providing information and early signals on root cases of failure. This will typically involve the collection and analysis of reliability data from many diverse sources at all phases of product design, development, manufacturing as well as use in the field. At the design stage of the product, the reliability goals are determined at the product (system) level, and then "flowed down" to the subsystem, component and sub-component level. The current design is assessed, "benchmarked", and compared to the new design where applicable. Gaps between the reliability goals for the new design and the current design are identified. In this phase, a major thrust is to identify the weaknesses of the new design and eliminate them. This stage makes use of a number of reliability engineering tools such as failure mode effects 693

694

R. Agrawal and N. Doganaksoy

analysis (FMEA), highly accelerated life testing (HALT), highly accelerated stress and strain testing (HASS) and failure reporting and corrective action system (FRACAS). O'Connor and Patrick (1991) provide useful background on these broader considerations for reliability improvement. Also see Meeker and Hamada (1995) for types of reliability testing and appropriate data sources throughout product development and field use. Once the product has passed the design stage, prototype units are built and tested to assess the reliability level. After the internal qualification of the product, prototype units are then taken out to the field for testing. Often, issues can be found in the field due to circumstances that the product designers were unable to anticipate, and therefore were unable to create in a laboratory environment. These field tests will most likely detect issues that are apparent in the early life of the product, since they often cannot be run for long, due to resource constraints. We know of one situation in which cockroaches in Alabama were accessing the circuit board of a product, and causing short circuits. This was not a situation that had been anticipated in the internal tests for the product. If no major issues appear in the field test then the product is ready for launch and regular manufacturing. In the situations that we have encountered, the design is intended to be "fixed" by this time. However, usually continuous on-going minor changes are being made to the product over time, either in an attempt to improve the product or because of cost constraints. We will return to this point later. Once the product is released for field use, there are usually two primary sources of data for monitoring reliability. • In-house audit test data: Manufactured product is sampled regularly and

subjected to some kind of internal accelerated life test, in an audit lab. • Field data: In an industrial context, we have found that field data can be generally classified into two periods of interest: the in-warranty period and the longterm life of the product. The reason for making this distinction is that usually the amount of information available in the two periods is substantially different. Meeker and Escobar (1998), Lawless (1982), and Nelson (1982) are useful references on statistical analysis of various types of reliability and lifetime data. In this chapter, we will discuss the use of the mean cumulative function (MCF) as a tool to analyze audit lab data. The outline of the chapter is as follows: We first describe the role of audit testing in reliability. We then present a particular case study that we were involved in and explain why we felt a more thorough analysis of the data was beneficial; we describe the M C F technique and we then show how it applies to the case study. Finally, we give some general observations and recommendations.

2. The role of audit testing in reliability

Reliability evaluations present a ¢hallenge beyond that normally encountered in quality evaluations because there is usually an elapsed time between when the

Analysis of reliability data from in-house audit laboratory testing

695

product is built and when the reliability information from the field is forthcoming. If the elapsed time is long compared to the manufacturing cycle of the product, a large number of units may be produced before it is possible to react to a reliability issue. Therefore, an earlier warning of potential reliability issues is desirable. An audit lab is used in a manufacturing situation to continually assess the quality and reliability of product being produced. Often, a small number of units are sampled from production and subjected to extensive testing. The units are then observed to determine how well they withstand testing. This cycle of sampling and testing is done with regular frequency, and significant changes in results may be an early indication of changes in the quality (and the reliability) of the product being manufactured. If failures and/of unexpected behavior of the product occur in the audit lab, then this can be carefully studied by design and manufacturing engineers to understand root causes. This is an important function of such a lab. To häve an in-house audit lab it is usually necessary to be äble to accelerate the life of the product in the laboratory, at an acceleration rate that is approximately known. This will make it possible to obtain reliability information in the lab at a fastet rate than in the field. One way of doing this is to cycle the product laster then we would expect it to be cycled in the field. Another way would be to subject the product to elevated levels of stress, for example high temperature or high humidity. Of course, it is always dangerous to assume that we can understand and reproduce the in-use conditions that a product observes in the laboratory. There taust, however, be a reasonable level of comfort that every attempt has been made to reproduce those factors that have a significant effect on the product life. Nelson (1990) and Meeker and Escobar (1998) provide in-depth discussion of accelerated testing and associated considerations. Why would we be interested in monitoring reliability at the audit lab? A valid question to ask hefe is, "Isn't it too late to be assessing reliability at the audit lab stage?" Clearly, if this was the only place in the product life cycle where we were assessing reliability, then the answer to that question would be yes. As part of a comprehensive reliability effort, however, it is important to recognize that the audit lab is our last defense before we find field problems as they are occurring in the field. Such problems may be a result of manufacturing issues, or because of ' minor" design changes that may have been caused by a change of supplier, raw materials, or other modifications. Judicious use of audit lab data could potentially mitigate millions of dollars in product service and replacement costs, not to mention the higher toll that might be paid in consumer dissatisfaction and loss of brand name, due to a major field issue. Further, there are also potential positive scenarios that can be capitalized on. While a monitoring tool can trigger a flag if the situation has changed for the worst, it can also trigger a flag if the situation has significantly improved, which might cause us to drill down further to capture and maintain the improvement. Such an improvement might otherwise have been missed. We have seen many situations in which audit labs can be used for the purpose described above.

696


(1) As in the case study of this chapter, an appliance manufacturer can use an audit lab on its toasters to monitor reliability. In this particular instance, accelerating the life of the product is relatively easy. Since a regular consumer might only use his toaster a couple of times a day, exercising the toaster almost continuously could quickly simulate a long period of life. Caution would be required in this case to not induce artificial failure modes - say, for example, in the overheating of coils due to constant use. (2) Manufacturers of power generation equipment are concerned with the reliability of dielectric insulation used on generator armature bars. This is a fairly mature technology and the nominal life expectancy of such insulation is on the order of several decades. However, unexpected changes in manufacturing or raw material conditions can result in premature failure of the insulation. In order to provide early notification of such problems, a small number of bars are sampled from production periodically - typically one or two bars a week due to relatively low volume of production. These bars are pur on a high stress voltage endurance life test to accelerate their failure times. Failure time data (often censored data since there are unfailed units) from different production periods (say, quarterly) are compared to detect changes in failure time distributions. (3) Thermoplastic resin manufacturers are concerned about changes in the color of automotive exterior body panels, molded from their product. Sample parts, molded from resin sampled during different production periods, are placed under highly intense UV light to detect possible changes or degradation of color.

3. Case study background We look now at a manufacturing process producing "toasters". We were primarity interested in the first year life of the toaster, since that was the warranty period of the product. We were also interested in the ten-year life of the product, since that was roughly how long consumers expect their toasters to last. An audit lab had existed at the toaster manufacturer's for some time, and there was a data collection system in place. The sampling scheine was such that five toasters were sampled randomly each week (typically in a systematic manner) from the manufacturing line, and put on "one year test". Also, approximately one toaster a week was taken and put on "ten year test". The reason for the discrepancy between units on one-year test and on ten-year test was simply a resource issue. Since these toasters were being manufactured in large quantity (about 80,000 toasters a month), this sampling represented a negligible amount of the overall population. The major costs were associated with testing and measurement rather than the cost of the units. The audit lab tests that were done were simply automatic rapid cycling tests of the toasters, allowing for sufficient cooling. Finally, we had some data that told us that on average, consumers use their toasters 400 times a year. One-year life of the toaster could be simulated in the lab in about two to three weeks.


697

I f a failure occurred on the toaster during its testing period, the component that caused the failure was repaired or replaced and the testing was continued. In this case, eight different failure modes (i.e., failing components) were observed on the toaster, the eighth category being "other". For the discussion that follows, it is assumed that a failure in a certain category always implied that the same component was replaced in the toaster. Although this was not always the case, it is a reasonable assumption for the categories of failure that were not "other". A segment of data gathered during a particular audit period is shown in Table 1. Even though there was a total of 30 toasters in this group, the table displays repair data from five units. Toaster T140 failed at 200 cycles (due to failure mode 3), at which point the failing component was repaired (or replaced). The unit was retired from testing after 425 cycles. Toaster Tl41 completed 410 cycles without any failures at which time it was taken oft test. Prior to our involvement in the audit lab, the data that resulted from the rapid cycling tests were used mainly for engineering analysis of the failure modes observed during testing. When a component on a toaster failed, the engineer responsible for that component was notified. Time permitting, the engineer would inspect the part and in some cases, do a root cause analysis. The test results were also plotted: the percent defect level vs the m o n t h of manufacture. This was also done by particular failure mode. It was felt that the analysis of this data could be strengthened. The largest issue was that the effect of age was not being accounted for in the current analysis. This is due to the fact that failing components were replaced with new ones. The failure rate estimates described above were based on number of toasters on test and did not take the age of the components into consideration. It was impossible to tell, therefore, whether there appeared to be an increasing failure rate, or a decreasing failure rate. Clearly, the former case would cause serious concern, while the latter case may be dealt with differently. Also, no quantification was being given to the variability in the data. Without such a quantification, results could not highlight

Table 1 Sample data. A failure mode of 0 here indicates that the unit was taken oft test Toaster number

Cycles

Fail mode

T140 Tl40 T141 T142 T143 T144 T144 T144 T144 T144

200 425 410 400 390 350 580 793 30t2 4011

3 0 0 0 0 1 6 7 1 0

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R. Agrawaland N. Doganaksoy

significant changes compared either to other manufacturing periods or to an absolute standard. Further, the tradeoffs in precision made between the one-year and the ten-year tests were not being conveyed. We saw an opportunity to make better use of this data, and the resources that are being put into the audit lab to generate it. In the next section, we discuss a non-parametric estimate of M C F as a monitoring tool for this data.

4. Using the mean cumulative function as a monitoring tool

4.1. Types of reIiability data The two common types of reliability (failure time) data are: (1) The time of failure for non-repairable units or components. Since a nonrepairable component can fail only once, time to failure data from a sample of non-repairable components consist of the times to first failure for each component. In most instances involving non-repairable components, the assumption of independent and identically distributed failure times is a reasonable one and suitable lifetime distributions (such as the Weibull or lognormal) are used to describe the distribution of failure times. Meeker and Escobar (1998), Lawless (1982), Nelson (1982) are useful references on statistical analysis of various types of reliability and lifetime data. (2) A sequence of reported system repair times for a collection of repairable systems. Data typically consist of multiple repair times on the same system since a repairable system can be placed back in service after repair. In some cases one can analyze time between repair times for a repairable system using analysis methods devised for non-repairable components. However, this approach makes the important assumption that after a failure of a system component, the act of repair (e.g., replacing the failing component) restores the system, with respect to that failure mode, to as good as new. When a single component or subsystem in a larger system is repaired or replaced after a failure, the distribution of the time to the next system repair will depend on both the overall state of the systems at the time just before the repair and the nature of the repair. Thus, repairable system data, in many situations, should be described with models that allow for changes in the state of the system over time or for dependencies between repairs over time. Since the toasters were being treated as repairable systems in the audit lab, the standard tool that is often used for reliability data, (e.g., fitting of a Weibull or lognormal distribution to times between failures), would not be directly applicable. 4.2. Mean cumulative function Repairable system data are viewed as a sequence of repair times. At a particular age t, each population unit has accumulated a number of repairs. One can

Analysis of reliability datafrom in-house audit laboratory testing

699

envision a population distribution of the cumulative number of repairs at age t, since different systems accumulate different number of repairs by age t. The distribution at age t has a mean M(t), the population M C F for the number of repairs. For a large population, M(t) increases smoothly with age t. It is useful to regard M(t) as a continuous function assumed to have a derivative ra(t) = dM(t)/dt. Here, ra(t) is the mean rate at which the number of repairs accumulated at age t. For the number of repairs, m(t) is called the "instantaneous repair rate function". It is also called the "recurrence rate" or "intensity function" if some other type of recurring event is observed. It is expressed in repairs per unit time per system, for example repairs per month per toaster.

4.3. Estirnation of mean cumulative function The model for such recurrence data is sometimes called a point process. Parametric methods used to analyze repairable system data assume a parametric form for M C F (of instantaneous repair rate function). See Meeker and Escobar (1998), Tobias and Trindade (1995), Engelhardt (1995), and Ascher and Feingold (1984) for discussion of parametric models for repairable systems. We will use a non-parametric approach to estimate MCF. Given a collection of repairable systems, a simple estimator of the M C F at time t would be the sample mean of the available cumulative number of system repairs for the systems still operating at time t. This estimator is appropriate if all the systems are still operating at time t (i.e., no censoring in the data). For example, for the data in Table 1, the estimate of M C F at 200 cycles is 1/5 (assuming there are only these five units in the group). Likewise, at 350 cycles, the M C F is 2/5. However, the estimation of M C F at 580 cycles is not as clear cut since the number ofunits is no longer five at this time. The data are multiply censored as a result of units retiring at different ages. An unbiased estimate of M C F allowing for multiple censoring is described in Nelson (1995), Nelson and Doganaksoy (1989), Meeker and Escobar (1998), and Tobias and Trindade (1995). These references also provide a discussion of assumptions underlying the non-parametric estimate of M C F and associated confidence bounds. Doganaksoy and Nelson (1998) extended this method to compare MCFs of two samples from two different populations of systems.

5. Application to the case study The analyses described in this section were done using SAS (1997) RELIABILITY P R O C E D U R E which is part of the QC software. The MCF, and associated confidence bounds, for three different periods of manufacturing, in chronological order are shown in Figure 1. The x-axis on these plots represents the number of cycles that the toaster has seen. The y-axis is the average number of repairs (or replacements). All three plots have been put on the same scale, for the purposes of comparison.


700

Period 2

Period 1 5 4.5 4 3.5 3

õ

] Upper95% CB ~

f ~ - -

~~

2.5

õ

5 4.5 4 3.5 3 2.5

2

2 1,5

1 0,5 500 1000 1500 2000 2500 3000 3500 4000 4500 CYCLES

1,5 1 0.5 0

«

500 1000 1500 2000 2500 3000 3500 4000 4500 CYCLES

Equivalent10 yearlife

Period 3

I

5

4.~ 3.5

2.~ 1.5

°5

~~

---__~__2 0 500 1000 1500 2000 2500 3000 3500 4600 4500 CYCLES

Fig.

1. M C F

estimates for three different manufacturing periods.

A preliminary look at this data appears to indicate that period 1 has a higher number of replacements than the other two periods. We also notice the tightness of the confidence intervals for the first 400 cycles, relative to those bounds for the remaining life of the toaster. This is a consequence of the sampling scheme discussed earlier. Let us assume that we are considering the manufacturing period 1 as the baseline for all subsequent manufacturing periods. Then, as we were testing period 2 manufacturing, we could be comparing it to period 1 by looking at the difference in the two periods. This plot is shown in Figure 2. Since the 95% confidence bounds for about the first 400 cycles of life are negative and do not include zero, we can conclude that manufacturing period 1 performs better in the warranty period than period 2. If we were doing this comparison as we were testing period 2 manufacturing, this might cause us to drill down further to particular modes, to understand where we are doing worst, and why. Note also that Figure 2 indicates that for the period of life when the toaster has seen more than 2000 cycles (a five-year equivalent), period 2 is better than period 1. Thus, we have improved the long-term reliability of the toaster. In between 400 and 2000 cycles, there is no apparent difference between the manufacturing periods. Similar comparisons can be done between period 1 and period 3, and manufacturing periods 2 and 3. These are shown in Figures 3 and 4, respectively. In Figure 3, we see that for about the first 1600 cycles (four-year equivalent), there is no apparent difference between the two manufacturing periods. After that, however, it seems that period 1 is worst. In Figure 4, we see that period 2 is


701

I Upper95%CB I

2500

0

3000

I Lower95% CB CYCLES

Fig. 2. MCF differences between period 1 and period 2.

/

2000

2500

3000

3500

4000

4500

5000

CYCLES

Fig. 3. MCF differences between period 1 and period 3.

slightly worst than period 3 for about the first 900 cycles, but then there appears to be no difference. One of the attractive features of M C F is that it can be applied in the same way to particular failure modes of the system, not just to the total number of replacements. When doing this, we ignore replacements due to failure modes other than the one of interest. The implicit assumption here is that replacing another component in no way affects the life of the failure mode of interest to us. We have

702


Æ

CYCLES

Fig. 4. MCF differencesbetween period 2 and period 3. plotted the M C F s for one particular failure mode of interest for each of the three different manufacturing periods in Figure 5. These are plotted on the same scale as in Figure 1, to allow an understanding of how much of the total replacement rate is due to this failure mode. Differences between the manufacturing periods are shown in Figure 6. There appears to be very little difference with respect to this mode in periods 2 and 3 - both appear better than period 1 in the mid-tolong-term life of the toaster. We can see, then, how these analyses allow us to monitor reliability in-house, and give us an early warning of potential field issues. This is especially true for those issues that might appear later in the life of the product. The audit lab information can signal a change in long-term reliability in this case within weeks of the actual manufactured time. This allows a much faster response than if we wait to discover the issue when our customers are experiencing them in the field. By using the audit lab data as an early warning indicator, we can potentially avoid or mitigate field issues. The potential impact of this system are reduction in the dollars spent on field issues, as well as an overall increase in customer satisfaction and loyalty to brand name.

6. Observations, recommendations and conclusions We have encountered a few issues when trying to implement the reliability monitoring system described here. We discuss some of these issues briefly.

Analysis of reliability data from in-house audit laboratory testing Period 1

703

Period 2

5 4.5 4 3.5

5 4.5 4

3 2.5 2

3 2.5. 2 1.5

3.5.

1.5 1 0.5 0 500

1000 1500 2000 2500 3000 3500 4000 4500 5000 CYCLES

0.5 ~ ..... ~ ..... 0 ,~ ---Y ........ T- - - ~ - - " " ~ ~ ' . 0 500 1000 1500 2000 2 5 0 0 3 0 0 0 3 5 0 0 4000 4500 5000 CYCLES

Period 3 5 4.5 4

8 = ®

3.5 3

2.5 2

0.5 0 500

1000 1500 2000 2500 3000 3500 4000 4500 5000 CYCLES

Fig. 5. M C F estimate for a particular failure mode for three manufacturing periods.

Period 1 and 3

Period 1 and 2 5 4

8

a

u.

~oo

o ;"%~'--t~'Sg00"~õ06

2500 30i10 3500 4000 4500 5000

-1

CYCLES

CYCLES

Period 2 and 3

8 ®

-õ u.

f ~- 5 " õ ~ ~ 4 0 0 0

4500 5000

CYCLES

Fig. 6. M C F differeuces for a particular failure mode.

704


The first obstacle we encountered when trying to implement this system was in justifying the M C F as the appropriate methodology. The people who were interested in this data were engineers, for the most part, who had some familiarity with reliability. They were acquainted, for example, with Weibull plots and standard mean time to failure calculations for an exponential model. There was some reluctance to embrace a new tool that was not generally familiar. The second issue we encountered was an organizational one, related to the fact that there were numerous people who had a stake in the audit lab results. It was owned by the people in manufacturing, and the focus there was on the in-warranty short-term issues. The people in engineering, however, were focusing on improving reliability and were in large part the customers of this early warning reliability system. Thus, it was difficult for us to achieve buy-in on changing the current system, since the people who would have most benefited from such a change were not the same people who could directly influence it. Another issue we encountered is that it was perceived that working on the audit lab was not going to make a large impact on the reliability of our products. It was felt that energy was better spent in the upfront design of the product. Again, we emphasize the importance of a complete reliability program that encompasses all stages of the product life cycle. Finally, this system is only as good as the data that go in to it, and we found that while the current data collection system was serving the current needs well, some changes could be made to it to improve the applicability of the results. We have discussed how the M C F can be used as a reliability monitoring tool for in-house testing. Such an early warning system allows us to understand when significant changes have occurred in a product's life. Especially for issues that occur later in life, these early warnings allow us to dramatically reduce our response time, thereby averting potential field issues that may have a negative effect on customer satisfaction.

Acknowledgements We would like to thank Gerry Hahn, Ed McInerney, Mike Ali, Jim Meyer, Paul Raymont, Todd Heydt and Carl Peterson for valuable assistance in doing the work that was described as the case study for this chapter.

References Ascher, H. and H. Feingold (1984). Repairable Systems Reliability: Modeling, Inference, Misconceptions, and Their Causes. Marcel Dekker, New York. Doganaksoy, N. and W. Nelson (1998). A method to compare two samples of recurrence data. Life Data Anal. 4, 51-63. Engelhardt, M. (1995). Models and analyses for the reliability of a single repairable system. In Recent Advances in Life-Testing and Reliability, pp. 79-106 (Ed., N. Balakrishnan). CRC Press, Boca Raton, FL.

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Lawless, J. F. (1982). Statistical Models and Methodsfor Lifetime Data. Wiley, New York. Meeker, W. Q. and M. Hamada (1995). Staöstical tools for the rapid development and evaluation of high-reliability products. IEEE Trans. Reliab. 44(2), 187-198. Meeker, W. Q. and L. A. Escobar (1998). Statistical Methodsfor Reliability Data. Wiley, New York. Nelson, W. (1982). Applied Life Data Analysis. Wiley, New York. Nelson, W. (1990). Aceelerated Testing: Statistical Models, Test Plans, and Data Analysis. Wiley, New York. Nelson, W. (1995)~ Confidence limits for recurrence data - applied to cost or number of product repairs. Teehnometries 37, 147-157. Nelson, W. and N. Doganaksoy (1989). A Computer Program for an Estimate and Confidence Limits for the Mean Cumulative Function for Cost or Number of Repairs of RepairabIe Systems. GE CRD Technical Information Series, 89CRD239, Schenectady, New York. O'Connor and D. T. Patrick (1991). Practical Reliability Engineering, 3rd edn. Wiley, New York. SAS (1997). SAS/QC Software: Changes and Enhancements for Release 6.12. SAS Inst., SAS Campus Drive, Cary, North Carolina. Tobias, P. A. and D. C. Trindade (1995). Applied Reliability, 2nd edn. Chapman & Hall/CRC, New York.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 200t Elsevier Science B.V. All rights reserved.

~')

z.ù K.y

Software Reliability Modeling, Estimation and Analysis M. Xie and G. Y. Hong

1. Introduction

Our society has become increasingly dependent on computer systems. Many system failures occur due to latent software defects encountered as the software executes various input combinations during operation. Software failure process is a random process and a systematic approach is needed to predict, measure and manage software failures so that the reliability of software can be quantified and improved. Software reliability is defined as the probability of failure-free operation of a computer program for a specified time in a specified environment. Software reliability parameters, measures and metrics, described by software reliability models (SRM), offer the possibility of evaluating and monitoring software failures quantitatively during the verification phases of a product. The probabilistic models for software failures and their frequency of occurrence can be described and measured by software reliability functions and mean time between software failures. In this chapter, a number of important software reliability models are reviewed. The focus is on the presentation of some typical models of different type with the emphasis on failure process modeling and reliability analysis. The review is not exhaustive as there are a number of published works reviewing the history of software reliability modeling, for example, Shooman (1984), Musa et al. (1987), Xie (1991), and Lyu (1996). Xie (1993) provides an annotated bibliography of 100 significant papers in software reliability and interested readers are also referred to the individual papers for detailed description of each model.

2. Classification of software reliability models

Many different models have been proposed, modified and adapted in the past few decades. The understanding of their differences and interrelationships can help the user to select appropriate model for different applications. The model clas707

708

M. Xie and G. Y. Hong

sification scheme is very important since it can give a clear picture of the existing software reliability models, signifies the relationships among different models, and aids the efficiency of model comparisons. Some existing model classification schemes are discussed in this section. Ramamoorthy and Bastani (1982) classified software reliability models into four types according to the phase of the software development: debugging phase, validation phase, operational phase and maintenance phase models. The authors also proposed another classification scheme according to the debugging strategy where software reliability models can therefore be divided into software reliability growth models, sampling models and seeding models. These schemes are appropriate for software engineers and developers so that suitable data are collected for model application. According to the nature of the failure process, Goel (1985) classified the existing approaches into four models: time between failure models, failure count models, fault seeding models and input domain-based models. This classification scheme is useful for practitioners as it clearly indicates the type of data needed for the analysis. Xie (1991) classified the software reliability models according to their probabilistic assumptions. Under this scheine, the existing models were grouped as: Markov model, nonhomogeneous Poisson process (NHPP) model, Bayesian models, statistical data analysis methods, input domain-based models, seeding and tagging models and software metrics models. This classification scheine will be adopted here. Nonhomogeneous Poisson process models a r e a group of models that has received most attention by researchers and practitioners. It is this type of models that will be emphasized here. Markov models contain a number of earlier models in the development of software reliability growth models and they will also be summarized. A number of Bayesian approaches to modeling and data analysis will also be mentioned separately. In fact, an important task in software reliability analysis is reliability prediction and data analysis, and they will be covered in a separate section. According to different criteria, software reliability models are classified into different categories. However, for a good software reliability model, it should have several important characteristics as follows: (1) it gives good prediction of future failure behavior, (2) computes useful quantities, (3) is simple to use, (4) is widely applicable, (5) is based on sound assumptions. It is well known that no single model will work in all situations and alternative models are usually needed. 3. Some important NHPP software reliability models

3.1. Concept of NHPP Denote by N(t) the number of failures occurred by time t. The process {N(t);t >_O} can be modeled by a NHPP. Because software reliability is in-

Software reliability modeling, estimation and analysis

709

creasing during the testing phase, this type of models is also commonly called software reliability growth model (SRGM). The general assumption for S R G M described by an N H P P is as follows: 1. 2. 3. 4.

A software system is subject to failures at r a n d o m caused by software faults. There are no failures experienced at the time t = 0. N o new faults are introduced during the testing. The probability that a failure will occur in an interval At is 2 At + o(At), where 2 is the failure intensity, which may depend on t. 5. The probability that more than one failure will occur in an interval At is o(At).

The function 2 (t) is called the failure intensity function and it is usually easier to work with the so-called mean value function given by

/0 t 2(s)ds

#(t) =

.

(1)

Note that #(t) represents the expected number of failures that are experienced at time t, i.e., #(t) = EIN(t)]. Divide the time 0 to t into a number of intervals, to = 0, q , . . . , ti-1, t l , . . . , tk = t. The number of faults in the /th interval, Ni, is a Poisson random variables with the mean E(Ni) = #(ti) - #(ti 1). N H P P models can be classified into two gronps: finite failure models and infinite failure models. The commonly known G o e l - O k u m o t o and S-shaped models belong to the finite failure model family. M u s a - O k u m o t o , Duane and logpower models can be classified into infinite failure family. As these models are representative and also widely used, we will give more emphasis to these models in the following. Software reliability models contain a number of parameters and they are usually unknown. To estimate the parameters of these models, the m a x i m u m likelihood method can be applied. Given the nnmber of failures in each interval, nj, we have that

P(Ni = ni) = {#(tl) - #(ti-1) } ni exp{-[#(ti) - #(ti-1)]} . ni! For model parameter estimation, m a x i m u m likelihood method can be applied. The likelihood function is given by

Models [NHPPSoftwareReliability E

1

I

[ FinitëFN,ureModels 1

I InfiniteFailureModels

T - -

.___[__

I

I

[ Goel-Okumoto [---S--Shaped ] [-Musa-Okumotõ] r L Model Modet , ~odo, J ,

I

Duane Model

Fig. 1. Some common NHPP models.

II

Log-power Model


710

k

L ( n l , . . . , nk) = I ~

{#(ti)

-

# ( t l _ l ) } ni

exp{-[#(t/) - #(ti-l)l}

(2)

ni!

i=1

By taking logarithm of both sides of equation, we have

--#--(ti-1)]niexp[-(#(ti) - #(ti 1))]

lnL = ~-~ln i=1

v

nj!

k =

~{n

i In[#

(tl) -- # (tl-l)]

- - E#

(ti)

-

#

(ti-1)]

-

In nil } .

i=1

The above formula can be simplified to k

ln L

= ~~~_~{niln[# (ti) - # (t,-l)] - In ni!} - #(&) . i=1

In general, to find out the maximum likelihood estimates, we can take the derivative of this equation and equate it to zero. It is also useful to obtain the confidence interval for the estimated parameters. Assuming that there are two parameters, a and b, we can use the Fisher information matrix to obtain their confidence intervals. The asymptotic variances Var(ä) and Var(/~), the covariance Cov(ä,/~) of the MLEs of the parameters are also needed for the calculation which can be derived following the standard a p p r o a c h . The Fisher information matrix is given by

I-E[ô21nL/~ 2 a I F = [_E[ô21nL/ôaôb ]

-E[ô21nL/aaôb] l -E[ô21nL/ô 2 bi

The asymptotic covariance matrix V of the M L estimators for parameters a and b is the inverse of the Fisher information matrix: V = F -1 =

[Var(ä) LCov(ä,/~)

Cov(ä,b)] Var(b)

(3)

The two-sided approximate 100a o'/o confidence limits for the parameters a and bare The above conclusions should be:

au = ä + Z~/2

V~ßär((~))

»u = b + z ~ / 2 ~

aL =

ä - Z~/2 ~

,

eL = b - z ~ / 2 ~ ,

where Z~/2 is the (1 - c~/2) quartile of the standard normal distribution. Given the mean value function #(t), the software reliability R(x[t) is defined as the probability of a failure-free operation of a computer software for a specified time interval (t, t + x] in a specified environment. Given the mean value function #(t), we have that

R(x[t) = e x p [ - { # (t + x ) - # (t)}] .

(4)

Software reliability modeling, eslimation and analysis

711

It should be noted that the above is testing reliability concept, and when mission reliability after the release is concerned. The proper concept is the operational reliability and this was recently pointed out in Yang and Xie (2000). The testing reliability concept is adopted here as illustration. With (4), the asymptotic variance Var(R) can be determined as

?R~ 2

Var(/~) = , e R , 2 Var(ä) +

+

Var(/~)

2(~R'~( ~R) a=a,»=äC o v ( ä , ~ ) k,8aJ ~

,

The two-sided approximate 100Œ% confidence limits for R is then given by

Rv=Æ+Z~/2~)

and

R L=Æ-Z«/2v/Var(R)

Similarly, the asymptotic variance for the failure intensity estimate is given by Var(2) =

ä(8)L'2~ Var(ä) ÷ ()2 \~aßa= ~62 b=$ Var(/~)

The two-sided approximate 100e% confidence limits for the true value 2 is

2u = 2 + Z~/2[Var(2 )] 1/2 and

)oL = 2-Z~/2[Var(~)] 1/2 •

3.2. Descriptions of specific models 3.2.1. The Goel-Okumoto model Goel and Okumoto (1979) developed a software reliability model, also called exponential growth model. It is characterized by the following mean value function #(t)=a(1-e

-bt)

a>0,

b>0

,

where a = # (oc) is the number of inherent faults and b indicates the failure occurrence rate. The failure intensity function 2 (t) is il(t) = abe -»t . A typical plot of the p(t) and 2(t) for the Goel-Okumoto model is shown in Figure 2. The ML estimates are the solution of the following equations:

712

M. Xie and G. Y. Hong 50

220

45 40

170

35

£ 30 O) -_=

120 1

.__=20

70 >

k1_

25 t-

ii O

15

20

10 5 0

-30 0

2

4

6

8 10 12 14 16 Time t (hour)

18 20

Fig. 2. A typical plot of the # (t) and 2 (t) for the Goel Okumoto model. {

k ^ ä = i~=lni/(1 - e-bt») , k

-

k

^

S-'( "' -21=~"------2');t.e-~t~ - t i _ l e-bti-1) = 0 \ e z;',-1-e -l;', 1-e-";~'k )t, The negative second partial derivatives of the lol likelihood function can be derived using the M L estimators ä and b of the G o e l - O k u m o t o model as follows: k

(_ô2 lnL/ßS a)== a = Z

nilä2 '

j=l

k (-521nL/52b)b=[~ = Z n i

i=1

( t i - ti 1)2e D(t~+ti 1) ^ (e-bti i - e &/)2

~_.2_-Dtk ätôe-Dt0 atkc +

and ^

^

(__~2 lnL/Sa~b)~=a,»=D = tke -btk - toe bto The software reliability R(xlt ) at the time interval (t, t + x] can be derived as R (x]t) -- exp[ae bt(e-óX - 1)l . The asymptotic variance Var(R) is obtained as Var(/)) = e 2ae ~"(e D~_ 1) × {e-2l;ti (e-& _ 1)•Var(ä) + ä2 [(tl + z)e -;(ti+~) - tie -[~t*]2Var(;) + 2äe -;t~ (e -& - 1)[tie [~ti_ (tl + z)e-b(ti+~))Cov(ä,/~) } •


713

The two-sided approximate 100«% confidence limits for R can also be derived. Similarly, Var(,~) is obtained by using the maximum likelihood and local estimates of the variances and covariance: Var(,~) = b2e-2&Var(ä) + ä2e-2&(1 - bt)2Var(/;) + 2äße 2/;t(1 -- t ~ t ) C o v ( ä , / ~ ) . The two-sided approximate 100«% confidence limits for the true value of failure intensity )~ can be derived. 3.2.2. The S-shaped models Yamada and Osaki (1984) discovered that some faults may be covered by other faults and cannot be detected at the beginning of testing. A model that can deal with this case is the delayed S-shaped N H P P model which has the mean value function #(t)=a(1-(l+ót)e-ót),

b>0

,

(6)

where a is the total number of faults eventually to be detected and b is a faultdetection rate parameter. The failure intensity function is 2(t) - d#Ds(t) -- ab2te-ót dt A typical plot of the mean value function/~(t) and failure intensity function 2(t) for the S-shaped N H P P model is shown in Figure 3. The MLEs of the parameter a and b can be derived as the solutions of the following two equations: 30

220

25

170 t-0

20

"6

12o

g

IJ _

70 "

10

20

'

~

'

'

'

2

4

6

8

10

~

'

12 14

'

'

-30

16 18 20

Time t (hour)

Fig. 3. A typical plot of the # (t) and 2 (t) for the S-shaped N H P P mode1.


714

ä=

~~=1 n~ (1 - (1 + Dtk)e -[~t~'

ù_~(

~~('~e-~"-t~-~e-~~~~) ,, ) = ä t Z e il + ~,t,-_~)e-»',-, - (1 + bt~)e

b~~

~',/

The software reliability R(x]t) at the time interval (t, t + x] can be derived as

R(x[t) = exp{ae-bt[(1 + bt + bx)e -bx - (1 + bt)]} . The asymptotic variance Var(R) is derived as

?R , 2

Var(Æ) =

+ \aa/

Var(ä) +

~

?R~ 2

Var(/;)

a=a,»=~

The asymptotic variance for the failure rate function is Var(2) =/;4fle 2btVar(ä) + ä2/~2t2e-Sbt(2 -- bt)2Var(/;) + 2ä/;3t2e 2b(2 -/~t)Cov(ä,/~) .

3.2.3. The Musa-Okumoto model Considering the possibility of infinite number of faults in the software, Musa and Okumoto (1984) proposed another NHPP model called the logarithmic Poisson model. It has a mean value function that contains two parameters /~(t) = aln(1 + bt) .

(7)

This lnodel reflects that faults with larger size are found earlier and it provides good results in modeling many software failure data. The failure intensity function is derived as

ab

;~(0 =

1 + bt

A typical plot of the mean value function and failure intensity function for the Musa-Okumoto model is shown in Figure 4. The maximum likelihood estimates of the parameters a and bare the solutions of the following equations: ä=

k ln(l+/~tk) '

k

li~l ll[~ti b =

ktk

(l+btk) ln(l+b&)

--0.

The software reliability R(x]t) at the time interval (t, t + x] is

Software reliability modeling, estimation and analysis 50

350

45 >., c c

715

300

40

cO

35

250 õ E

30

200 ti-

25 150

20

e100 0~

15 10

/

5

FailureIntensity X (t)

0 2

i

i

i

i

i

i

i

4

6

8

10

12

14

16

Time t

50 i

18 20

(hour)

Fig. 4. A typical plot of the # (t) and 2 (t) for the Musa Okumoto model.

~1~ R(xlt) = [1 +l+bt b(t + x Then the asymptotic variance Var(R) is derived as

1 +b(t + x ) J

Var(ä)

1 +~+x)

ä(l+/~t)2~-2 Var(/~)_2{ B [1 +[~(t+x)] 2~+2

l+/~t

}a

1 +~(t+x)

~ln{1 -FffQ~-x) 1+~~}äl~+~~lat«ovlä,~ [1 +[~(t+x)] a~+l

The asymptotic variance of the failure intensity function can be obtained as follows: /~2

ä2

äß

Var(,~) - ¢ t .+~ (b) 1 Var(ä) -~ (1 + t~t)4 Var(b) + . t . ~+ ( b) 1 Cov(ä, b) .

3.2.4. The Duane model The Duane model was one of the earliest models proposed for hardware reliability (Duane 1964). Duane noticed that if the cumulative failure rate versus the cumulative testing time was plotted on log-log paper, it tends to follow a straight line. Later on, Crow and Singpurwalla (1974) observed that it can also be referred to as a Weibull process. It is an NHPP in which the failure intensity function has the same form as the hazard rate for a Weibull distribution. This model is sometimes called the power model since its mean value function for the cumulative number of failures by time t is taken as the power of t, which is


716

# ( t ) = a t b,

a>0,

b>0

.

(8)

For this model, the cumulative number of failures observed at time t will tend to be on a straight line on a log-log scale plot. The slope of the fitted line gives us an estimate of in a. This graphical interpretation provides us a simple way to estimate the parameters and it is very useful in reliability analysis (Donovan and Murphy, 1999). This model has the failure intensity function

),(t) = abt b-1 A typical plot of the mean value function #(t) and failure intensity function 2(t) for the Duane model is shown in Figure 5. The MLEs of the parameters a and b a r e of close form and they are given by (Crow and Singpurwalla, 1974) k,

ä

k

4

D

~~-j1 ln(&/ti)

The software reliability R(xlt ) at the time interval (t, t+ x] can be derived as

R(xlt) = exp[at b - a(t + x) bI . The asymptotic variance Var(R) is Var(/~)= { S R ) 2 V a r ( ä ) + (/aR'~ 2 Var(b)

\ ~ / «:ä

\ e b/»=»

~~'~(~~)

~ov~ä,~~

==a,b=~

+ kCa) ~

50 380

45 40

¢O

>~ 35

330 =

30

Lt.. 13)

_c 25

280

20 ij_

c-

15

IlJ

230

10

'

'

'

2

4

6

'

'

.

.

.

8 10 12 14 Time t (hour)

. 16

180 18 20

Fig. 5. A typical plot of the/x (t) and )~(t) for the Duane mode1


717

Then the asymptotic variance for the failure rate function is Var(~)

b2t2/;-2Var(ä) + [ät~-1 + äb(D - 1)tb-2]2Var(b)

=

+ 2ä/~t2~-2[1 + / ~ ( b - 1)t-l]Cov(ä, b) .

3.2.5. The log-power model Modifying the Duane mode1, Xie and Zhao (1993) proposed an N H P P model called log-power model. It has the mean value function #(t)=alnb(l+t),

a>0,

b>0.

(9)

An important property is that the log-power model has its graphical interpretation. If we take the logarithmic on both sides of the mean value function, we have ln#(t) = lna + b l n l n ( 1 + t) . If the cumulative number of failures is plotted versus the running time, the plot should rend to be on a straight line on a log-log scale. A first-model-validationthen-parameter-estimation approach is also discussed in Xie and Zhao (1993). This model has the following failure intensity function ,~(t) :

ab in b-~ (1 + t) l+t

A typical plot of the mean value function/~(t) and failure intensity function B(t) for the log-power model is shown in Figure 6.

30

280

Mean Value function

25

~t (0

260 240 = O

B 20

22o o~

~-15

2oo g

lt.

180 ~160 ~ 140 0

i

i

i

3

5

7

i

9

i

I

i

i

120

11 13 15 17 19 21

Time t (hour) Fig. 6, A typical plot of the #(t) and )~(t) for the log-power model.


718

The MLEs of the parameters a and b a r e ä-

k lnb(1 + t k ) '

b=

k

klnln(1 +tk)

- 2~=~ lnln(1 +tl)

The software reliability R(x)) at the time interval (t, t + x] can be derived as

e(xl«)

= exp{alnb(1 + t) - alnb(1 + t + x ) } .

Then the asymptotic variance Var(R) can be calculated using the following equation: Var(Æ) = , 2

Var(ä) +

\8a /la=ä

÷

Var(/;)

\ 8 b J D=D

2//~"' ~( 8") Cov(~,~;) ~Ôaß ~ b a=ä,b=[,

.

Then the asymptotic variance for the failure rate function is Var(i) =

;2 ln2~-2(1 + t) Var(ä) (1 + t) 2

+

ä2 W»-2(~ + t) [~ + blnln(1 + t)]2Var(/~) (l + t) 2

÷

2ä/;ln2;-2(1 + t)[1 + »lnln(1 + t)]Cov(ä,b) . (1 + t) a

3.3. A case study N H P P models are widely used by practitioners. Usually point estimates are used and there are many examples available. An example is given in this section to illustrate the applications of the interval estimation of parameters and reliability prediction. A piece of software was developed and then tested for 28 weeks. The complete failure data were recorded and given in Table 1. Table l Number of failures per week from a large communication system Week

Failures

Week

Failures

Week

Failures

Week

Failures

1 2 3 4 5 6 7

3 3 38 19 12 13 26

8 9 10 11 12 13 14

32 8 8 11 14 7 7

15 16 17 18 19 20 21

7 0 2 3 2 5 2

22 23 24 25 26 27 28

3 4 1 2 1 0 1


The Goel-Okumoto

719

m o d e l is u s e d h e r e . T h e M L e s t i m a t i o n s b e c o m e s t a b l e

only when an adequate amount of data are accumulated. The MLE results of the l a s t f e w w e e k s a r e l i s t e d i n T a b l e 2. The 95% confidence intervals for parameters a and bare calculated using the m e t h o d i n t r o d u c e d i n t h e p r e v i o u s s e c t i o n s a n d t h e r e s u l t s a r e l i s t e d i n T a b l e 3. The 95% confidence intervals for the software failure intensity and reliability p r e d i c t e d i n t h e f o l l o w i n g w e e k s a r e l i s t e d i n T a b l e 4.

Table 2 The estimation of parameters a and b using ML methods Week

Failures

CMF

a

b

20 21 22 23 24 25 26 27 28

5 2 3 4 1 2 1 0 1

220 222 225 229 230 232 233 233 234

258.9 256.3 256.3 268.5 255.8 255.3 256.3 250.2 249.2

0.09472 0.09578 0.095597 0.083328 0.095575 0.095751 0.092254 0.099146 0.099855

Table 3 A 95% confidence intervals for parameters a and b Week

a

ac~

aL

b

bv

bL

20 21 22 23 24 25 26 27 28

258.9 256.3 256.3 268.5 255.8 255.3 256.3 250.2 249.2

224.7 222.6 222.8 233.7 222.7 222.5 223.4 218.1 217.3

293.2 290.0 289.8 303.3 288.9 288.2 289.2 282.3 281.1

0.09472 0.09578 0.09560 0.08333 0.09558 0.09575 0.09225 0.09915 0.09986

0.07397 0.07519 0.07544 0.06548 0.07626 0.07682 0.07407 0.08058 0.08159

0.1155 0.1164 0.1157 0.1012 0.1149 0,1147 0,1104 0,1177 0,1181

Table 4 A 95% confidence intervals for the failure intensity and software reliability predicted for the following weeks Week

)~

)~u

)~L

R

Ru

RL

28 29 30 31

1.52 1.38 1.24 1.13

2.11 1,92 1,75 1,59

0.93 0,83 0.74 0.66

0.74 0.76 0.78 0.80

0.83 0.84 0.86 0.87

0.65 0.68 0.70 0.73


720

3.4. Some extensions The models described in the previous sections are simple and commonly used ones. However, they are based on a number of assumptions. Depending on actual situation, some assumptions should be relaxed and there a r e a number of N H P P models developed for specific applications. Considering software faults of different type, Yamada et al. (1985) developed two-type of failure. Pham (1996) extended it by considering three different error types: type 1 error (critical) - very ditficult to detect; type 2 error (major) difficult to detect; type 3 error (minor) - easy to detect. The model also allows for the introduction of any of these errors during the removal of an error, which is commonly known as imperfect debugging. The mean value function of the software reliability model is 3

#(t) = Z

api [1 - exp(-(1 - fii)bit)]

i=1 1 -/~i

where pi is the proportion of type i fault. Yamada et al. (1986) incorporated the concept of test effort in the GoelOkumoto model (Goel and Okumoto, 1979) for a better description of the failure phenomenon. Later, Xia et al. (1992) incorporated the concept of a learning factor in the same model. They have considered the effects of test effort and learning process to describe the failure process independently, but in reality, test effort and learning process are dependent on each other. Chatterjee et al. (1997) presented a software reliability growth model which incorporates the joint effect of test effort and learning factor. The overall mean value function has the following form:

#(t)=a

1-exp

f 2kb

-bLl" ~ 2kb2a ) In

~ c~

~ J=+pt kb2 "

where a, b a r e the same notations as in the Goel-Okumoto model, k is proportionality constant, and c~, fi are constants. The parameters a, b, cq/3 and k can be estimated by using the traditional maximum likelihood method. Although the parameter estimation based on N H P P models can be carried out in a standard way, Knafl and Morgan (1996) presented the results of general twoparameter N H P P model. Zhao and Xie (1996) discussed the problem of parameter estimation in more details. Model comparison and validation is another interesting statistical problem. In most of the papers presenting a new model, some results comparing with other models are usually presented. Gaudoin (1998) studied the test for the Duane model and it can also be used for the log-power model.


721

4. Markov software reliability models Software failure process can be considered as a counting process and software reliability can be studied under the framework of Markov assumption. Markov models are very useful in studying software fault-removal processes. Especially, many earlier studies tend to focus on this type of models.

4.1. The Jelinski-Moranda (JM) model One of the earliest software reliability models is the Jelinski and Moranda (1972) model. The underlying assumptions of the JM-model are: (1) There a r e a fixed number of unknown initial faults within the software; (2) When a fault is detected, it is removed immediately and no new fault is introduced; (3) Times between failures are independent, exponentially distributed random quantities; (4) All remaining software faults contribute the same weighing factor to the software failure intensity. Denote by No the number of software faults in the software before the testing. The initial failure intensity is then equal to N0~b where q5 denotes the failure intensity contributed by each fault. Hence after removing the kth fault, there are (No - k ) faults left, and the failure intensity becomes ~b (No - k ) . Let ti be the observed time between the ( i - 1 ) s t and the /th failure. Then T/, i = 1 , 2 , . . , N 0 , are then exponentially distributed random variable with parameter

)~(i)=~(No-i+l),

i=l,2,..,N

0 .

The distribution of Tj is given by P(T/ 0} is given, the parameters q~ and No in the JM-model can easily be estimated by maximizing the likelihood function. The likelihood function of the parameters No and ~b is given by


722

L(tl, t 2 , . . , th; No, ¢) = 11 ¢(N0 - i + 1). exp[-¢(N0 - i + 1)ti] i=l

=q5 n

-i+l

.exp -

-i+l)ti

By taking the partial derivatives of this log-likelihood function with respect to No and ¢, respectively, and equating them to zero, we get the following likelihood equations: 8 lnL_~ 1 ~ No i=1 No --i + 1 lnL -n ~ ê~b

~i=1 (N0

~--~~q~ti = 0, i=1

1 -i+1

) ti=O .

Usually numerical procedures have to be used to solve these two equations. However, the equation system can be simplified as follows. By solving q5 from the second equation above we ger -1

B=n

-i+1

ti

and by inserting this into the first equation, we obtain an equation independent of ¢. The estimate of No can then be obtained. 4.2. A general DFI formulation

An assumption with the JM-model that is widely discussed is that all software faults are assumed to be of the same size. In fact, some faults are more easily detected than the others. A modification of the JM-model is presented in Xie (1990). Basically, the JM-model can be modified by using other intensity function 2(i). A failure intensity function 2(i) is said to be decreasing failure intensity (DFI) if 2(i) is a decreasing function of i. Software failure process under Markov assumption can be described as a Markov process model with DFI function. The theory for continuous time Markov chains can then be applied such as that the collection of probabilities {pi(t)=P[N(t)=iJ;

i=0,1,2,..,N0,

t_>0}

satisfies the so-called Kolmogorov's differential equations. The power-type DFI Markov model assumes that the failure intensity 2(i) is a power-type function of the number of remaining faults, that is 2 ( i ) = ¢ [ N o - ( i - 1 ) ] ~,

i= 1,2,..,N0

.


723

As we expect that 2(0 decreases fastet at the beginning, it is reasonable to assume that 2 ( 0 is a convex function of i. Hence, ct is likely to be greater than one. Another type of function that satisfies this requirement is the exponential-type Markov DFI model. It assumes that the failure intensity is an exponential function of the number of remaining faults. It is characterized by the failure intensity function 2(0:-qS[exp{-fi(N0-i+l)}-l],

i= 1,2,..,N0 .

The parameters in 2(0 can be estimated by maximizing the likelihood function given by L(t, 2(.)) = ~

2(0 e x p { - 2 ( i ) t i }

.

i=l

4.3.

The Shanthikumar

general Markov

model

The JM-model has been generalized by using a general time-dependent transition probability function by Shanthikumar (1981). This model assumes that the failure intensity function as then number of faults removed is given by 2(n, t) = ~o(t)(No - n) , where qS(t) is a proportionality factor. Under the Markov assumption, the forward Kolmogorov's differential equations can easily be obtained and solved together with some standard boundary conditions. This model reduces to the JM-model when ~b(t) = 1. Furthermore, a special case of this model is known as the Schick-Wolverton model (Schick and Wolverton, 1978). It has a linearly increasing function, i.e. qS(t~) = cti. That is, for the Schick-Wolverton model, the failure intensity function after detecting the /th fault is 2(ti) = O(No - i + 1)tl . Note that the failure intensity function of the Schick-Wolverton model depends both on i, the number of removed faults and ti, the time since the removal of last fault. The probability distribution function of N ( t ) is denoted by Pù(t). Under the Markov assumption, we have that the forward Kolmogorov's differential equations are given as follows,

aPo(t) ~t

--

-No~(t)Po(t),

8Pn(t) _ (No - n -F 1)~)(t)Pn_l(t) - (No - n)~)(t)Pn(t); at

1 < n < No •


724

Using the boundary conditions P,(0)= 0, n > 0 and P0(0) = 1, this system of differential equations can easily be solved and the solution is given by

P~(t)=(N°)[a(t)]N°-'[1-a(t)]';

l 0, for some constant A. This means that if N and A are known, then T/has an exponential density, f ( t i l 2 i ) = 2ie -xit~ ,

(2)

where 2i = A ( N - i + 1). Some modifications of JM model (1) are presented in the literature. Moranda (1975), supposed that the fixing of bugs that cause early failures in the system reduces the failure rate more than the fixing of bugs that occur later, because these early bugs are more likely to be the bigger ones. Thus, Moranda (1975) assume that the failure rate should remain constant for each T/, but that it should be made to decrease geometrically in i after each failure, that is, for constants D and K, rr, (tID, K ) = D K i-1 ,

(3)

wheret>0,D>0and0 0, fij > 0, 0 < Oj < 1 and 01 + 02 = 1, but not including covariates. Different choices for ~.j(t),j = 1,2 in (26) could be considered. As a special case, consider a Musa and O k u m o t o (1984) with intensity function ) q ( t ) = o~/(fll ÷ t) and an exponential process with intensity function

B2(t) = fi2t. Assuming a failure truncated model, the likelihood function for ~, fll, fi2, 7 and is (from (12)) given by,

( ~-[((~

O~

0

fl2x;02i))

×exp(--Oln~log(l+~l)-O2n~2x~2-),

(27)

where 01n

eV+~wn 1 + e~+~w~ '

02n

1 - 01~ .

Assume the following prior distributions for c~,fil, fi2, 7, and,

~~r(a~,bl), B1 ~'~ß(a2,62), (28)

f12 ~'~ß(a3,b3),

7NN(dl,cl), ~NN(d2,c2)

,

where al, aa, a3, bi, b2, b3, cl, c2 are known and N(#, er2) denotes a normal distribution with mean # and variance «2. Also assume prior independence among the parameters. Also assuming the introduction of latent variables Ii = (Iil, Ii2), where Iij = 1 if the ith failure is caused by t h e j t h type and//j = 0 otherwise, j = 1,2, i = 1 , . . . , n,

Bayesian analysis for software reliability data

743

we obtain the joint posterior distribution for O~,fl,f2,7 given by

and r (see (21)),

(Ui~~t=I Oll)(Hi~Ii2=lX*02i) " f 1 q7 X*) Hi:Iil=l(

X ca

~,=1

Pl

P2

c

×exp(-(bl-kOInl°g(l+~l))C~

-- (b3 + ×exp(

(?-&)2 2«1

X;2\

(z-d2)2) 2-«7 -J '

(29)

where

O~i-

e?+Zwi 1 + e~ +~w, '

02i = 1 -- Oli .

The conditional distributions for the Gibbs algorithm are given by (i) Construct I given c~,fl, f2, 7, z and 9 by generating independent variables//1 from the Bernoulli distribution with parameter O~Oli Pil ~ ~Oli -~- f202ix*(fl + x*)

where/i2 = 1 -//1 (ii)

=(~lfl, f» 7, ~,z, 9)

F al +

~~1;bi+ 01nlog 1 +

.

i=1

(iii) ~(f2[ ~, fit, 7, ~,I, 9)

B

a3 -t- n --

(30)

/11; b3 qi=1

(iv) 7z(flIŒ, f2,7, z,I, 9 ) 0< fl~2-1e b2flll]/l(0~,fl,f2,7,

T) ,

744

J. A. Achcar

where

(

(~1) _ Z"i i ~

01 (c~,fil, 82, 7, r) = exp --O~Olnlog 1 + x•

log(81 -t- xf)

i=1

)

(v) (

(ù~ -- dl) 2

~(WIc~,S1,82, z,I,=@) oc exp

("c - d2)2"~ , , 2-c2 ") ~'/j2('~,81' 82' ')2»27)'

2Cl

where B2( ~, 81,82, 7, T) = exp

(~

Iil

log Oli

J[i2log 02i

-~-

\i=1

-01nc~ log 1 +

i=1 *

02

Xn _

Observe that we need to use the Metropolis-Hastings algorithm to generate the variables 81,7 and z. Similar results could be obtained considering more than one covariate w.

6. Bayesian inference and model determination

We can use the Gibbs samples to ger inferences on the parameters of the software reliability model or on functions of these parameters. In this case, we could approximate posterior moments of interest. As a special case, consider the mean value function ra(t). A Bayes estimator of ra(t) with respect to the squared error loss function is given by the posterior mean E(m(t)]~). Similar inferences could be obtained for 2(t). For model selection, we could consider (see Raftery, 1996) the marginal likelihood of the whole data set N for a model M, given by

v. 1 the minimum variance occurred at r~ - r3/2. The value r~ was slightly less than r3/2 for values of 3 closer to 1 and increased monotonically to about r3/2 as 3 increased. The asymptotic variance ~ S changed very little for r2 values near r~. For ~ _< 1 it appeared that better results (in terms of smaller variance) would be obtained with smaller values of r2. However, for larger values of r3, values of r2 close to rl = 1 result in ~ ( 3 ) - 0, which is of no use in estimating 3.

Direct graphical estimation for the parameters in a three-parameter Weibull distribution

757

As a compromise, we tried using r2 = Lr3/4J when 6 _< 1. Unfortunately, while this procedure resulted in reasonable estimates for 6, using r2 = ~r3/4J sometimes produced terrible estimates for/?. This led us to consider using only r2 = Lr3/2J. As a check on how well the asymptotic variance predicted behavior for small n, on how the bias in the estimation of 6 behaved, and on how robust the choice of r2 = ~3/2J was to different 6's; we simulated the expected value and variance for 6. We generated Weib(g, 1,0) samples of size n = 20,30, and 50 for values of 6 representing both increasing and decreasing hazard rates. R a n d o m numbers were generated using the S U N performance library routines, simulations were done in F O R T R A N , and the samples were simulated 500 times for each combination of 6 and n. The value of rl was fixed at 1; different amounts of censoring corresponding to r3 =0.5n, O.67n, O.75n,n were used; and both r2 = lr3/2J a n d r 2 = ~r3/4J were studied. For each sample the statistic T was calculated and 6 was obtained using Eq. (2.9) for all the r3,r2 combinations. The sample means and the sample variances of these estimates of 6 are given in Tables A1-A7. These tables suggested that using just r2 = ~r3/2j would be reasonable.

4. A graphical solution While it would be possible to create a computer program to perform the estimation procedure described above, out interest was in providing a procedure that required only a few graphs. The first sets of graphs (Figures 1-4) are graphs of B ( 6 ) and are used to obtain 6. The four figures correspond to values of r3 = 0.5n, 0.67n, 0.75n, n. Estimates of/~ are obtained using Figures 5 8. Again, the figures correspond to values of r3 = 0.5n, 0.67n, 0.75n, n. Given n, r3, and 6 (r2 is a function of r3); the figures provide the denominator in Eq. (2.10). Finally, Figure 9 provides the coefficient of ~ i n Eq. (2.11), from which Nis obtained. EXAMPLE 4.1. Reanalyzing the data in Example 2.1 graphically, one would proceed as follows. In this example rl = 1 and r3 = n = 20, and therefore, one would use r2 = L20/2J = 10. As in E xample 2.1, ones computes T = 0.4053 and from Figure 4 the estimate of 6 is 6 = 1.5. F r o m Figure 8 (with 6"= 1.5) one obtains the denominator for Eq. (2.10) as 0.63 and ».

B-

O.4O2 O.265 - 0.217 . 0.63

Finally, from Figure 9 one obtains the coefficient for ~" in Eq. (2.11) as 0.12 and ~ ' = 0.265 - 0.217(0.12) = 0.239 .

758

P. R. Nelson and K. B. Kulasekera

2.5

2.0

1.5 LI-

I

~~~I~!t I111t 1

ù,,

g:,.--.

/Q;

1.0 -

I'. ,'."" 1.O

0.5

i i

,/ 0.0

i

I

2

1

0

3

4

5

6

Fig. 1. Graph of ~ ( 6 ) for P3 = 0.5n and c~E [0.5, 7].

~I~I~!!/~

2.5

2.0

/ ù ,, :.";.

1.5

.

.

.

.

.

.

, ' la.

)~I" ....... ù ."

1.0

B.,"l"

!~ ~ I

I

"-" "---'-

0:5

/.7' 0.0

i

i

I 0

1

2

3

Fig. 2. Graph of ~ ( 6 ) for r 3

4

5

0.67n and 6 c I0.5, 7].

6

7


2.0

- -

nlO n20 n30 n50 n lO0 I

1.5

/ "::;ù--'12.... ......

L tl.

1.0

! ù: 0.5

0.0

0

I 1

t 2

I 3

I 4

5

6

Fig. 3. G r a p h of ~-(&) for r3 = 0.75n and & E [0.5, 7].

1.2

0.8

0.4

0.0

0

1

2

3

4

5

6

Fig. 4. G r a p h of ~(&) for r3 = n and & C [0.5, 7].

6

7

759

760


nlO n20

/,

n30 n50

0.4

/ P

/ .,/'" ..........

nlO0!

Ot

v

o

¥

s' , , , . . . . . . . . . . . . . '" . . . . .

ù,i/:/,' J'*s"

0.3

,f «

0.2-

:/

0.1 -

'

i.ùæ..

'',.

f

f

0.0 -

2

3

4

5

6

7

G

Fig. 5. G r a p h of the denominator in Eq. (2.10) for estimating beta when r3 = 0.5n and ~ E [0.5, 7].

0.5 ------

nlO n20 n30 n50 nlO0 ~°.**,

¢- 0.4 0

,

,,

J

I

',,,, 0,3

--

ù",,,,,,, ",,,,,,

¥

f

v

0.2 -

/

0.1 -

2

:3

4

5

6

G

Fig. 6. Graph of the denominator in Eq. (2.10) for estimating beta when r3 = 0.67n and ~ c [0.5, 7].


o6!

I

761

I nlO n20

- - ~ ........

J

n30 n50 , nlO0

.,

/.'i

L*"

"*%

0.5

.~.

,~.~..

,rt-

*%% .%

O I

ù,

~'~*.% ~.%. '%'**°

~ ~~.~.

'",

~ 0.4-

+ ~" o.3 ", I_: 0.2

2

3

4

5

6

7

5

Fig. 7. G r a p h of the denominator in Eq. (2.10) for estimating beta when r3 = 0.75n and 6 c I0.5, 7].

I

I

/

"%%

0.7 -

'% '%~. ",%. ù , ' '%~ *%*% ,% '%%% %*%. ~'~. ,~

~~~,'........ .,% ~-

0.6-

(J

,-~ c

ù,

\

0.5

o

~v

'%~ *'**% ,% =%**

,,

%'~

\

0.4 nlO n20 n30 n50 nlO0

•~ -

0.3 --~

0.2

I 0

1

2

3

4

5

6

7

B

Fig. 8. G r a p h of the denominator in Eq. (2.10) for estimating beta when r 3 = n and 6 E [0.5, 7].


762

J 0.6 -

- -

~

I ......... I .....

I.....

I

nlO

"2°1 n301

/

nS°l

ù,

/

'-- 0.4 /

v

¥ 0.2

""'/ i 0.0

2

3

4

5

6

7

Fig. 9. G r a p h of the coefficient for # in Eq. (2.11) for estimating ~/when 6 c [0.5, 7].

5. Comparison with other graphical procedures

5.1. Censored data The only other graphical procedure that works with right-censored data is a Weibull probability plot, so we compared our procedure (referred to as N & K in Tables 3-6) with the Weibull probability paper developed by Nelson (1967) (referred to as LSN in Tables 3 6). For the comparison we generated values from a Weib(6, 1, 0) distribution, and used 1000 trials for each 6, n, r3 combination. The results are summarized in Tables 3-5. Obviously, it was not feasible to plot all the generated values by hand to obtain estimates using Weibull probability paper, so we wrote a program to do that estimation for us. Recall that when plotting data on Weibull probability paper, one taust first shift the data so that y - 0 by subtracting an estimate of y from each data value. This is a trial and error procedure, and one would generally start by plotting the unshifted data to see if there was any indication (i.e., curvature) that y was not zero.

Out first attempt at this computerized estimation transformed the (x,y) coordinates of each point so that the transformed points would rend to fall on a straight line if the data were Weib(6, /3, 0) (this mimics what the scales for the x and y axes on the probability paper do), fitted a line to the transformed points using


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least squares, and computed the R 2 value associated with the line. The appropriate amount ~" to subtract was obtained iteratively by repeating the procedure, each time subtracting a slightly larger value for 7 and comparing the resulting R 2 values. If at the first step R 2 decreased, we assumed ~"= 0 was appropriate. Otherwise, the amount subtracted was increased until R 2 reached a m a x i m u m value. Using this computerized estimation led to some very bad estimates of 6, and we discovered that in m a n y instances (particularly when only a few points were being plotted) the values X(1) and/or X(2) seemed to unduly influence the slope of the fitted line. If one were actually fitting a line by hand, they would tend to compensate for this by allowing these first two points to fall further from the line. As a counterpart to that we had the program check to see if deleting X(1) and refitting the line resulted in an estimate of X(1) that was too far away from the actual value (more than 25~). I f it was, the point X(1) was deleted. IfX(1) by itself did not have undo influence, the program checked the pair )((1) and X(2) by deleting them both, fitting a least squares line to the remaining points, and checking if either value was more than 25ä from its predicted value. I f either point was too rar away from the fitted line, they were both deleted. Before running the simulation we checked this procedure for deleting values by comparing it with simply using the least squares line obtained without deleting any small observations. We found that deleting values lead to estimates that were uniformly less biased and had smaller variances. Even using the deletion procedure there were still a few cases where the Weibull plot produced very large estimates for 6. We decided that when the estimate of 6 exceeded 8, to just use 8 as the estimate. The rational for this was that one could extrapolate from the scale on the Weibull paper to some degree, but at some point one would simply have to decide that 6 was "very large" (i.e., 8). The number of times this happened was recorded. Thus, there were three different situations where something in Weibull plot estimation procedure had to be modified. They are referred to as faults in Tables 3-6. The occurrences of these faults are recorded as ( f l , f 2 , f 3 ) , and Table 7 indicates which type of fault each count corresponds to. Table 7 Explanation of the fault counts in Tables 3-6 Fault count

Situation

f~ f2 f3

deleted X(1) and )((2) were deleted

ä-s.o X'(1 ) was

Examination ofTables 3 5 suggests the following. Out äoverestimates a and has a larger variance than the Weibull plot, which underestimates õ. In m a n y situations it would be preferable to overestimate a, thus inferring a larger hazard rate. The two estimators of fi are rauch closer. For a close to 1 our estimator appears better in terms of both bias and variance. This is particularly true for 6 = 0.5 and r3 _< 50, where the Weibull plot does a poor job in terms of both bias and variance. The smaller r3 is, the worse the results are. However, as ô increases from 1,

772


the situation reverses and the Weibull plot estimates have both smaller bias and variance. The Weibull plot estimate of 7 appears to have both smaller bias and smaller variance, p r o b a b l y due to it not depending on the estimates ~'and fi as ours does. Our estimate, however, does not appear unreasonable. Also, our estimators are easier to obtain since there is no trial and error plotting involved. 5.2.

U n c e n s o r e d data

Table 6 compares our procedure with both the Weibull probability plot and the procedure o f C o h e n et al. (1984) (referred to as C W D ) . The comparison of our procedure and the Weibull plot is similar to what was f o u n d in the censored comparison. Our procedure and that o f C W D behave similarly. In some circumstances one has smaller bias and variance, and for other circumstances it is reversed. Both procedures tend to overestimate c5. Our procedure is worse in this regard for large Ô and large n, while C W D is worse for ~ - 0.5 a n d / o r small n.

6. Conclusion We have presented estimates for the three parameters o f a Weibull distribution that are based on three order statistics and can be obtained graphically with relative ease. They work with either complete samples or with right-censored samples. F o r rightcensored data our procedure compares favorably with the Weibull probability plot and is simpler to implement. It does, however, tend to overestimate the shape parameter 6. F o r complete data it is competitive with the procedure o f C o h e n et al. (1984); but works best with small samples. F o r larger samples, the estimates o f C o h e n et al. (1984) have both less bias and smaller variance.

Appendix Table A1 Estimates and their variances when ~ t'/

t"3

0.5

?'2 = [r3/2j

r2 = it3/4j

x

,2(x)

x

s2(x)

20

20 15 13 10

0.5418 0.5408 0.5554 0.6313

0.1647 0.2219 0.2837 0.5392

0.5125 0.4933 0.5069 0.5093

0.I255 0.1962 0.2481 0.4411

30

30 23 20 15

0.5183 0.4781 0.5249 0.5533

0.1513 0.2044 0.1969 0.2930

0.5425 0.4450 0.4967 0.4979

0.1230 0.1628 0.1755 0.I984

50

50 38 33 25

0.5281 0.4787 0.5104 0.4984

0.1410 0.1584 0.1712 0.1422

0.5021 0.4684 0.4818 0.4855

0.1029 0.1140 0.1276 0.1292

Direct graphical estimation for the parameters in a three-parameter Weibull distribution Table A2 Estimates and their variances when c5 = 0.75 n

r3

r2 -

Lr3/2j

r2-

Lr3/4j

;-

,2l~

a-

,2(x)

20

20 15 13 10

0.8378 0.8525 0.7047 0.9672

0.3207 0.4664 0.4787 0.8213

0.7864 0.7608 0.7948 0.8134

0.2725 0.3297 0.4452 0.7201

30

30 23 20 15

0.5024 0.7717 0.8166 0.8429

0.3773 0.3032 0.3126 0.5271

0.6899 0.7370 0.7661 0.7540

0.3898 0.2297 0.2625 0.3541

50

50 38 33 25

0.7732 0.7525 0.7737 0.7596

0.1712 0.2105 0.2006 0.2389

0.7736 0.7608 0.7402 0.7451

0.1309 0.3297 0.1619 0.2100

Table A3 Estimates and their variances when 6 = 0.8 n

rs

r2 - Lrs/2J

r2 - Lrs/4J

20

20 15 13 10

0.8971 0.9174 0.9858 0.9994

0.3554 0.5194 0.7275 0.8695

0.8415 0.8197 0.8593 0.8842

0.3005 0.3735 0.4996 0.7919

30

30 23 20 15

0.9639 0.8280 0.8758 0.9057

0.4101 0.3290 0.3473 0.5712

0.7012 0.7939 0.8198 0.8118

0.3698 0.2458 0.2926 0.4049

50

50 38 33 25

0.8520 0.8161 0.8267 0.8120

0.1379 0.2125 0.2167 0.2488

0.8373 0.7468 0.7902 0.7960

0.1309 0.2036 0.1730 0.2391

773

774


Table A4 Estimates and their variances when 6 = 0.9 n

r3

r2

=

Lr3/2j

r2 = [ r 3 / 4 ]

s

,2(~-~

x

,2(x)

20

20 15 13 10

0.9918 1.0521 1.0121 1.1019

0.4348 0.6224 0.8054 0.9597

0.9514 0.9393 0.9941 1.0012

0.3441 0.4635 0.6269 0.8898

30

30 23 20 15

0.8636 0.9304 0.9981 1.0036

0.4580 0.4022 0.4271 0.6594

0.8723 0.9006 0.9296 0.9295

0.3077 0.2898 0.3646 0.5101

50

50 38 33 25

0.9434 0.9158 0.9307 0.9163

0.1723 0.2499 0.2556 0.2959

0.8447 0.8297 0.8872 0.8972

0.2618 0.2375 0.1992 0.2891

Table A5 Estimates and their variances when ~ = 1.1 ~

r 2 - [r3/4j

~ = L~/2j

3"

~2(g'/

3"

s2(;'t

20

20 15 13 10

1.2618 1.3133 1.3915 1.4851

0.5731 0.7737 0.9474 1.1372

1.1761 1.1901 1.2661 1.3031

0.4537 0.6696 0.8554 1.1351

30

30 23 20 15

1.1009 1.1671 1.2553 1.2886

0.4914 0.5542 0.6265 0.8183

1.0663 1.1177 1.1586 1.1809

0.3493 0.4166 0.5402 0.7168

50

50 38 33 25

1.1561 1.1351 1.1438 1.1295

0.2236 0.3070 0.3607 0.4266

1.0996 0.9989 1.0774 1.1024

0.1592 0.3202 0.2791 0.4340

Direct graphieal estimation for the parameters in a three-parameter Weibull distribution

775

Table A6 Estimates and their variances when 6 = 1.5 r2 = Lr3/4]

n

r3

ra -

Lr3/2]

20

20 15 13 10

1.7482 1.8234 1.8906 1.9704

0.8091 1.0303 1.1635 1.3296

1.6366 1.6834 1.7402 1.7531

0.6980 1.0055 1.1102 1.4122

30

30 23 20 15

1.5927 1.6735 1.7739 1.7865

0.5806 0.8187 0.9208 1.0732

1.5448 1.5754 1.6303 1.6985

0.4602 0.6897 0.7958 1.0805

50

50 38 33 25

1.5820 1.5811 1.5936 1.5852

0.3639 0.5267 0.5857 0.7104

1.4988 1.4397 1.4950 1.5468

0.2758 0.4815 0.5148 0.7043

Table A7 Estimates and their variances when ~ = 2 n

r3

r2 =

Lr3/2J

r2 - - [ r 3 / 4 J

x

s2(x~

x

s2(x)

20

20 15 13 10

2.3097 2.3531 2.3977 2.4159

1.0030 1.2075 1.3094 1.4364

2.1852 2.1880 1.2291 2.1040

0.9443 1.2203 1.2975 1.5402

30

30 23 20 15

2.0818 2.2684 2.3514 2.3125

0.8422 1.0662 1.1384 1.2427

2.1083 2.1524 2.2181 2.2318

0.6967 0.9755 1.0944 1.3572

50

50 38 33 25

2.1412 2.1583 2.1741 2.1411

0.5720 0.7913 0.8607 0.9733

2.0263 1.9898 2.0483 2.1150

0.4654 0.7325 0.8021 0.9989

References Cohen, A. C., B. J. Whitten and Y. Ding (1984). M odified m o m e n t estimation for the three-parameter Weibull distribution. J. Qual. Technol. 16, 159-167. Johnson, N. L., S. Kotz and N. Balakrishnan (1994). Continuous Univariate Distributions-1. Wiley, New York. Nelson, L. S. (1967). Weibull probability paper. Ind. Qual. Contro123 (11), 452-453. Nelson, W. (1972). Theory and applications of Hazard plotting for censored failure data. Technometrics 14, 945 966.

N. Balakrishnan and C. R. Rao, eds., Handbook of Statistics, Vol. 20 © 2001 EIsevier Science B.V. All rights reserved.

"~ ]

,)

Bayesian and Frequentist Methods in Change-Point Problems

Nader Ebrahimi and Sujit K. Ghosh

The change-point problem is one of the important problems of statistical inference in which one tries to detect abrupt change in a given sequence of random variables. This problem, which originally started with statistical control theory (see Page, 1955), has now been applied to different fields, including but not restricted to survival analysis and reliability studies. The literature about change-point problem, by now, is quite extensive. In this paper, our goal is to review recent developments in this area. In particular, statistical procedures to estimate discrete change point as well as continuous change point are reviewed.

1. Introduction

There are two types of change-point problems: (i) continuous change-point problem and (ii) discrete change-point problem. In the continuous change-point problem we assume that a continuous random variable T representing a survival time or a failure time has the hazard function

h(t) = ~ h°(t) i f O < t < t o ,

( hl(t)

ift>to

(1.1)

,

where ho(to) ¢ hl(t) as t --+ to. In other words h(.) has a discontinuity at to. The problem is to estimate the change point to. In the discrete change-point problem we assume that X 1 , . . , X n are observed as a sequence of independent random variables with an abrupt change at g0 E { 1 , .., n}. In other words, there exist two probability distribution functions F0 and FI such that e0 i~I Fl(xi) . P(X1 to) ,

(2.1)

where

I(A)= {1

ifxcA,

0

otherwise .

Using the n-ordered observations t(1) < t(2) _< ..- _< t(n), they construct a kernel X~(t) such that the solution of Xn(t) provides a consistent estimate of to. The construction of X~(t)is as follows. Define

BI(T,t) = T,

B2(T,t) = I(T > t),

B4(T,t) --- T2I(T > t),

B3(T,t) = TI(T > t),

B = (B1,B2,B3,B4),

Il

Bj(t)=lZBj(t(i),t), n

j=1,2,3,4

,

i=1

and _

±

+ (1 -

. . . . /~3 IJ2[t)) ~ (t) +/~l(t) log/~2(t) •

The kernel Xn (t) is then x . ( o ) = o,

Xn(t(i)) = H@(t(i)),

i=l,...,n-1,

xn(t)

t _> t(n -

= x,,(t(n

- 1)),

1)

.

The construction of Xn(t) is ingenious but apart from providing a consistent estimate this estimator does not seem to have any attractive properties. Later, Yao (1986) and Nguyen and Pham (1987, 1989) propose an alternative method for estimating to. It is clear that for each n, the likelihood function is Ln(20, )«, to) =

!~ i=1

log f(t(i)) =

Jo~

(log y(t))dFù(t)

,

(2.2)

N. Ebrahimi and S. K. Ghosh

780

where Fn is the empirical distribution function and f(t) is given by Eq. (2.1). Now the procedure to get estimator of to will be in the following three steps: (i) For each fixed to < t(n), maximize Ln with respect to 20 and 21; (ii) Insert the values of 2o and )~1 from the step (i) into Ln. Denote the resulting function by L*~(to) and maximize it by varying to; (iii) Insert the value of to from step (ii) and maximize the function Ln to get the new values of 2o and 21. Repeat step (ii) with the new values of 2o and 21 to get new value of to. Return to step (i) and continue the remaining steps until the algorithm converges. It should be noted that as to tends to t(n) from below, Ln tends to infinity. Thus, we are led to restrict L* (to) to some random intervals JAn,Bnl depending on data alone, with 0 _< An < Bn < t(n). Examples of An and Bn are given by Nguyen and Pham (1989). Note also that the function L~(to) is not defined at data points t ( 1 ) , . . , t(n), only its limits at to tend to these data points, from below or above exist. Thus, even by restricting to to [A~,B~], we may not be able to achieve a maximum in step (ii). In order to solve this problem one can maximize L*~*(to) =L;(t0) if t(r) < to < t(r+ 1),r = 0, 1 , . . , n - 1 and = max{L~(tö) , L*(tö-)} if to = t(r),r = 1 , . . , n 1, where t(0) = 0 and L;(to) and L~(t +) are the left- and right-hand limits at to, respectively. Matthews et al. (1985) also considered the model ho(t) = 20 and hl (t) = )q, but they were interested in testing the hypothesis H0 : 21 = 2o against Ha : 2o > 2~. See also Loader (1991). Basu et al. (1988) extended the model proposed by Nguyen et al. (1984) to allow general ho(t) keeping h2(t)=)~1. They suggested two semi-parametric estimates for to based on the assumption that the proportion of population that dies or fails at time to or before is known. Let pl be such that po < pl < l, where po is the known number of items in the population that failed at time to or before. Also let k be the number of order statistics between T([ùpo]) and T([npl]), and

B, = 2(t(i) log F~(t(i) ) /k + 1) - (~~))(~-~~ log l~~(t(i) ) /k + 1)) t2(i) __ (~-~ t(i) ~2 k+l \z-~ k+lJ

and the summations range over i = [npol + 1 to i = estimators by Basu et al. (1988) of to are t0(1) = Inf{t

[np~]. Then the proposed

: y~(t + h~) - yn(t) t ) rt '~Po

Bayesian and Frequentist methods in change-point problems

781

Basu et al. (1988) proved consistency of their estimators. Later Ghosh and Joshi (1992) investigated asymptotic distribution of t0(2). Modeling the aging process by the change-point mean residual life function was initiated by Ebrahimi (1991). He considered the model

m(t) = mo(t)I(O < t < to) + mlI(t > to) , where ra(t) is the mean residual life function or the remaining life expectancy function at age t and is defined as { ' f ö # ( x ) d x if~e(t) > 0, m(t) = E ( X - t]X> «) = P~(t) 0 if Æ(t) = 0 . He proposed an estimator for to and also studied its asymptotic properties.

2.2. Bayesian methods None of the classical methods described in Section 2.1 is quite satisfactory and puts stringent restrictions in order to obtain asymptotic normality. Moreover, simulation studies show that asymptotics are poor for small-to-moderate sample sizes. Bayesian approach on the other hand avoids asymptotics and provides more reliable inference conditional only upon the data actually observed. Howerer, Bayesian methods are also susceptible to impropriety of posterior distribution of to if one is not careful in specifying the prior for to. In fact, Ebrahimi et al. (1999) showed that an improper prior on to, necessarily leads to an improper posterior for to. Achkar and Bolfarine (1989) consider the model ho(t) = 2o and hl(t) = 21 and avoid this problem by using a discrete uniform prior for to. However, such a choice could tremendously limit the scope of application. A nice review on problems that arises within Bayesian framework is presented in Ghosh et al. (1993) which concentrates on the case 2o _> ,~1. In fact Ghosh et al. (1993) shows that one needs the restriction 2o >_ -~-1 for some known 2o in order to make the posterior proper. Ebrahimi et al. (1999) gave general Bayesian formulation of the change-point problem and they discussed the case of h ( t ) = 2o > h l ( t ) = 21 which yields particularly simple fitting and the Weibull case. Returning to (1.1), Ebrahimi et al. (1999) adopt fully parametric modeling assuming ho(t)= ho(t;Oo) and hl (t) = hl (t; 01). Thus, h0 and hl are two, possibly distinct, parametric families of hazard functions indexed by 00 and 01, where 00 and 01 could be vector valued. To capture the order restriction on h0 and hl they ler S = {(00, 01) : h0(t; 00) _> hl(t;O1) for all t > 0}. The likelihood takes the form n

L(Oo, 01, to; t) = I-I h(tj) exp(-H(tj))

,

(2.3)

j=l

where t = ( q , . . . , th) denotes the observed values of the lifetimes. Let vj(to) = 1 if tj < to, = 0 if tj > to. Then (2.3) becomes

N. Ebrahimi and S. K. Ghosh

782

L( Oo, 01» to; t) = I~I ho( tj; Oo)V/t°) hl ( tj; 01) 1-vj(t°) j--1

× exp

- H(H0(min(tj, to); 00) + [Hl(t£ 01) - Hl(t0; 01)]+) I(S) . j=l " (2.4)

For their model they restrict the likelihood so that to _> t0). Certainty of a change-point during the period of observation would then add the further restriction to < t(n). In order to complete probability specifications, Ebrahimi et al. (1999) require a prior distribution for 00, 01 and to. They assume that it takes the general form

f(Oo, 01)-f(S). f(to)

(2.5)

and that it is proper which assures that the posterior f(Oo, 01,t0]t) is proper. The prior information on to places it on the interval (0, b) with b possibly oc. The actual support for to is truncated according to the restrictions imposed by the likelihood. When to is not bounded above they argue that iff(t0) is improper the posterior must necessarily be improper. Combining (2.4) and (2.5) provides the complete Bayesian specification and thus the posterior f(Oo, 01, tolt) which is proportional to

L(Oo, 01, to; t). f(Oo, O1)f(S) . f(to) •

(2.6)

The posteriors f(00[t) and f(01lt) enable us to learn about the pre- and postthreshold hazards. In fact for each t, since h0(t; 00), H0(t; 00), h l (t; 01),//1 (t; 01), h(t; 00, 01, to) and H(t; 00, 01, to) are all random variables, they all have posterior distributions which would be of interest as well. However, primary interest is in the posterior for to,f(tolt) and when a change is not certain, P(to > t(n)It). The expression in (2.6) is not analytically tractable so they turn to simulationbased approaches for fitting such a model and use Markov chain Monte Carlo techniques. For more details see Ebrahimi et al. (1999). Even if the proposed models are all parametric, they can be quite rich if one incorporates finite mixture distributions. The semi-parametric approach is proposed in Ebrahimi and Ghosh (1999) where the problem is formulated in terms of dynamic weight mixture model. Also, the methodologies described by Ebrahimi et al. (1999) could be extended to detect more than one change point. However, one must be careful in specifying the joint prior distribution of change points.

3. Discrete change-point problem Discrete change-points problem occurs as a result of non-homogeneity in a sample. In some cases we have a priori knowledge about the physical nature of the process that generates data and we propose some parametric family to characterize such knowledge. However, one must construct a statistical test based

Bayesian and Frequentist methods in change-point problems

783

on initial data to check the validity of the model. In this situation a vicious circle arises: in order to create such a model one must guarantee the statistical homogeneity of data, but it is just the same model which is used to check the statistical homogeneity hypothesis of the data obtained. Thus non-parametric methods are the only satisfactory way to tackle this kind of problem. Between opposing parametric and non-parametric methods there is a large area of semi-parametric methods of change-point detection. The broad applicability of discrete change-point problem in various areas makes this area an attractive field of research. In this section we give two applications. For more applications see Zacks (1983) and Broemeling (1972). Suppose we are monitoring the rate of occurrence of a rare health event, for example a specific congenital malformation. Since the number of malformed births is small, one can assume that the malformed births occur according to a realization of a Poisson process with parameter say 21. Suppose an epidemic occurs at an unknown instant of time and the normal rate is subject to increase. (Environmental risk factors such as toxic spills, contaminated drinking water and radiation may also increase the normal rate.) Let V be this change. Since the interarrival times for a Poisson process are i.i.d, exponential, X1,X2,.. ,Xv will be independent having common exponential distribution with parameter 21 and X v + l , X v + 2 , . . will be independent having common exponential distribution with parameter 22, where 22 >_ 21. The goal is to estimate V. Another important application is on-line quality control of a manufacturing process. Imagine a machine that produces some product. The machine might break down at some point. The purpose of an on-line quality control scheme is to determine, based on the observation of the manufacturing process, whether the machine is functioning properly or not. In this setting, it is assumed that the observations are independent and their common distribution function before the change is F0 and after the change is F1. Clearly, F0 = F1 implies that the machine is functioning properly. In this section we discuss several Frequentist as well as Bayesian approaches to the discrete change-point problem.

3.1. Frequentist approach Assume that there exists 1 < g0 ~< n - 1 such that the joint probability density function of X ~ , . . , X n is ~0

I-[ f(Xi;O1) ~ I f(x"02) i 1

'

i=e0+ 1

where f is known and 01 ~ 02. If we assume that both 01 and 02 are known, then the maximum likelihood estimator of g0 is r

80 = a r g

max ~ ~

l 0, is the root t of BT(t ;3) = k + n * , or t~-~Z~[1]~,

k=0,1,...

We develop now the probability distribution of T(3). Let 2" = 2~/7~7211]/ô, and let

O(vS(k) =P2{r(3)

= tl} ,

k = 0,1,...

(2.10)

Since the Poisson process is strongly Markovian (see Cinlar, 1975, p. 117), the probability function {~(8 (k), k >_ O} can be determined recursively according to the equations ~~~) (0) = P (n*; 2* x/~7)

(2.11)

The operating characteristics of sequential procedures in reliability

793

and, for k _> 1,

#/*

l=0 k-1 j=l

(2.12) A non-iterative solution of (2.11) and (2.12) is not available. However, the computation of {tp(r;) (k), k _> 0} is very fast. Once the values of ~~) are determined one can immediately determine the moments of N(6) and of T(cS) and their quantiles. The results depend, however, on the selected initial sample size n*. We illustrate this dependence in Table 1, in which c5 = 0.1,7 = 0.9. We denote by Q1,Me and Q3 the first quartile, the median, and the third quartile of N(6), respectively. Generally, as expected, Eo{N(6)} is slightly increasing with n*, while //õ{N(cS)} is slightly decreasing. The quantiles Q1,Me, Q3 are quite robust against changes of n*. Furthermore, as 0 increases, the distribution ofN(~5) becomes more symmetric. Notice also that, according to Table 1, if

Os
O

.

Table 3 Expected value and variance of T(6), & = 0.1, y = 0.9

0

n*

Eo{T(6))

Vo{T(6)}

2 3 4 5 O

40 10 3 2 1

133.58 86.93 62.08 48.38 37.89

257.95 293.13 338.53 294.70 290.34

S. Zacks

796

Moreover, the coverage probability of

777'7 --X) is the same as that of

l÷5'l:B)

"

Thus, with such an experiment we can estimate p(O) sequentially with the desired proportional closeness probability but reduce the expected duration by a factor of m, since

Eo{ Tm(6) } = 1Eo{ TI (6) }

.

(2.23)

It is often the case that m independent systems are put on test, but failed systems are not replaced. In this case the interfailure times are independent but not identically distributed, and all formulae should be changed. We will still use the stopping boundary Bm(t) given by (2.20). Thus, 7],ù(6) defined as in (2.21) has the same possible stopping points, but the probability distribution of Nm(6) has to be changed. Let b(j'; n,p) and B(j; n,p) denote, respectively, the p.d.f, and the c.d.f, of the binomial distribution with parameters (;0 (n,p). Let Ob,m(k),k >_O, denote the probability function of Nm(6) when failed systems are not replaced. We obtain the following recursive equations: @(;0<m,(05,= B(n*; m, 1 - e where 2~, =

2*/m,

__0} is a Poisson counting process with intensity m#. We consider then the boundary B~ )(t) = -kl + rot, and the upper boundary B(Ü) (t) = k2. In this case the true points at which B ~ ) (t) can be crossed Table 5 Values of 2(#), E~{T}, and T05 for kl = 3, k2

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.9999 0.9989 0.9885 0.9406 0.8237 0.6456 0.4552 0.2974 0.1867 0.1159 0.0725 0.0459 0.0294 0.0192 0.0126 0.0084

42

6.00 7.42 9.31 11.65 14.87 t9.10 23.19 25.89 26.95 26.85 26.12 25.11 24.01 22.90 21.83 20.82

4.5 5.5 6.5 7.5 12.5 20.5 29.5 30.0 28.9 27.60 26.09 24.63 23.26 22.00 20.83 18.80


are (t}m),j = 0 , . . ,k2 - 1}, where t!m) = with the stopping boundaries B(m)(t) and

Ora.kl (~), ~;~ • J = Pu{Km(Tm) = j}

=

801

tj/m. If Tm denotes the stopping time B(m)(t)= k2, then

~,/.(~) l,kt

Ü)

for a l l j = 0 , . . ,kz - 1. Thus, the OC function is the same as in the case o f m = 1, but E~{Tm} = E~{T~}/m. If such an experiment can be performed a considerable saving can be attained in termination time. We consider now the case of m parallel systems, without replacement of failed systems. Let Jm,b(t) be the total number of failures in (0, t]. Consider the stopping time Tm,b = inf{t : Jm,b(t) = - k l 4-

rot,

or

Jm,b(t)

=

k2}

•

(3.18)

We define the crossing probabilities

(~)

O~l,m»(J)

"

=

Pu{-kl

+

mTm,b= j } ,

j = 0, 1 , . . , k2 - 1. One can determine recursively, t)(v) kt ,m,b ;0' \ ) = b(0;m, 1 -

e -k'~/m)

(3.19)

and for j _> 1,

(~)

~'k,,~»(J)

"

b(j;m,

=

1 -

e -~(k~+')/m)

j-I -

V/~a~Pkl,m,bk ' , (u) ;l'b(j-1; ]

-

m

-l,l-e

-~(j-O/m)

(3.20)

/=0

Also here the second term on the right-hand side of (3.20) is a convolution, and one can obtain explicit solutions to (3.19) and (3.20). However, for computational purposes, one can use Eqs. (3.19) and (3.20). The OC function is k2-1

~b,.,(/~)

(~) " • = Z O_O;G(t) = 1 - e ut, t > O, for w h i c h oo

P{Wa > t}

~p(j; j-0

g(a - t))P(j; At) .

(5.7)


807

Differentiation of (5.7) yields the p.d.f. (5.4). We conclude the present section with the following result. Let M(a) = E{W~}. From (5.5) we get for absolutely continuous F and G,

M(a)

= a(1 -

F(a)) +

/0~(1 - F(t))dt

+ ~~_lfoa (F*(n)(t) -F*(n+l)(t))G*(n)(a-t)dt .

(5.8)

Let f(t) and g(t) be the densities of F and G, respectively, and let fr(s) and be their LT's. Then, the LT of M(a) is

M*(s) =

g*(s)

e ~aM(a)da

(

1

B(1 - f*(s)) \1 + 1 -

1

"~ 1 d * s ,

f*(s)g*(s)/ + -s" dssf ( )

s>0

.

(5.9) In particular, in the double exponential case

1( M*(«)-«(Ä+,)

1+ (~+s)(~_+_s).~

~

(5.10)

Inversion of this LT yields

M(a)

(~2 + # )

~ - R + # + a e Xa

_ _+e -~)2 (;.+U)a (~

(5.11)

Recall that M(a)/a is the proportion of total time in (0, a] the system is ON. From (5.11) we obtain the well-known formula lim M(a) a--+oo a

_

#

2+#

(5.12)

Eq. (5.9) can yield the LT of M(a) for any distributions F and G. Inversion of this transform yields the formula of M(a).

6. Sequential detection of wearout

Generally, systems operate for long periods with an almost constant hazard rate function. This phase in the life of the system is called the "mature phase". After the mature phase comes the "wearout phase", in which the hazard rate is increasing. For purposes of controlling the operation of a system, the inventory of spare parts, etc., it is important to detect, as early as possible, the shift epoch from the mature phase to the wearout phase. Zacks (1984) modeled the problem as that of estimating the epoch of shift to the wearout phase, which is a parameter

s. Zacks

808

%0 < z < oo, intrinsic to the system. The hazard function of the system is modeled as

h(t;fl, Œ,z)=

{~ ~,+fl~Œ(t-z) ~-1

if t _< z, ift>~ ,

(6.1)

where (2, c~,z) are all positive parameters, and ct > 1. The shift of h(t; .) from a hazard function of an exponential to that of a Weibull distribution with ~ > 1 (increasing), is at the time point r. Moreover, the hazard at t > -c is greater than Ä. The life distribution of such a system has a density function

f(t;)~,c~,~) = fle-~t(1 + ~ ( 2 ( t - ~)+)~-1). e x p { _ 2 ~ ( t _ r)+} ,

(6.2)

where (t - z)+ = max(0, t - z). When z is unknown, one should be able to estimate it from the available data on failure times of such systems. We remark here that usually we consider this problem with repairable systems. Thus, time measurements on such systems are cumulative operation time. Zacks (1984) studied systems which are repaired immediately after failure, or after A units of operation time, for preventive maintenance. The data were the history of sequential random failure or maintenance times Tl, T 2 , . . , where 0 < T1 < T2 < ' ' " < Tn < " ' '. A Bayesian estimator of z, was derived. For the first time the estimate of z is smaller than the chronological time, the shift is detected. We do not present here the formula of the Bayesian estimator of r, which is not of a particular interest in a general discussion. Moreover, the detection rule mentioned is not necessarily optimal. For optimization one needs to formulate a risk function which depends on various loss components, like the loss, per time unit, of late or of early detection. Generally, the derivation of the optimal detection rule is difficult. A few exceptions are in cases of Markov decision processes, see Zacks (1991b) for details.

7. Sequential methods in software reliability Software reliability studies have special models, since the software is a finite collection of units (modules, functions, etc.), which does not change if a fault is not found and corrected. Several models of software reliability and sequential testing procedures are discussed in Zacks (1995). In the present section we will present some of the stopping rules, which are of special interest. Since the article of Zacks (1995) is a comprehensive chapter in a book, we will not give here all the details. Software might contain a certain unknown number, N, of faulty units. As long as no faulty unit is chosen for execution, the software fulfills its requirements. What is software reliability? The reliability function R(z) is the probability that the computer running with the given software will survive z time units without crashing or showing faults. There are two main software reliability models: time-domain models, and data-domain models. A common time-domain


809

model can be described in the following terms. Suppose there are N faults in the system where N is unknown. The model assumes that the random time required until a fault is detected by the requirement of a random customer is exponentially distributed. The times to detect different faults are i.i.d. Thus, the observed times of the first n detected faults are distributed like the first n-order statistics, T(I:N) < T(2:N) < "'" < T(n:N), from a sample of N i.i.d, r a n d o m variables. If we denote by E(fi) the exponential distribution with mean/~, 0 #i,

- - 0 0 < # i < OO,

(7i > 0

.

(1.1)

Although technically the Pi could be any real numbers, in the context of lifelength distributions they are positive and are referred to as the guaranteed lifetimes. To describe the basic goals and the background of the selection and ranking, we will first consider ranking the populations according to the values of the location parameters (guaranteed lifetimes), assuming that al . . . . . crk = « (known). It is also assumed that there is no prior knowledge about the correspondence between the ordered and the unordered #i. Most of the investigations in the literature are concerned with the goal of selecting the population associated with the largest #~ or the one associated with the smallest #» the target population being called the best population. Two classical formulations of these problems are known as the indifference zone (IZ) formulation due to Bechhofer (1954) and the subset selection (SS) formulation due mainly to Gupta (1956). Let us consider selecting the population associated with the largest #i. In the IZ approach, one of the k populations is selected as the best. A correct selection (CS) occurs if the selected population is a best population. Let P(CSIR) denote the probability of a correct selection (PCS) using the rule R. It is required that a valid rule R satisfies: P(CSIR ) _> P* whenever #[k] -

#[k-I] ~ 3" »

(1.2)

where the positive constant c5" and the guaranteed PCS P* (k -1 < P* < 1) are specified in advance by the experimenter. For a rule R based on a single sample of size n from each population, one has to determine the minimum sample size n for which (1.2) holds. 813

814

K. Hussein and S. Panchapakesan

The part of the parameter space defined by the restriction: #[k] - ~[k-1] ~ ~* is called the preference zone (PZ) and its complement is the so-called IZ. For selecting the population associated with the smallest #» the PZ is characterized by: BI2] -- //[1] --~ b*.

In the SS approach for selecting the best population, the goal is to select a nonempty subset of the k populations so that the selected subset includes the best population (which results in CS) with a guaranteed minimum probability P* whatever be the configuration of the unknown/~i- It is assumed that, in the case of a tie for the best population, one of the contenders is tagged as the best. The size S of the selected subset is not specified in advance but is determined by the data. The usual measures of the performance of a valid rule are the expected subset size E(S) and the expected number of non-best populations included in the selected subset (which is equal to E(S) - PCS). In order to meet the guaranteed minimum PCS under either formulation, one needs to evaluate the infimum of the PCS over the appropriate set of configurations of/~ = (#l,---,#k). Any configuration of # for which this infimum is attained is called a least favorable configuration (LFC). For selecting the best two-parameter exponential population where the best population is either the one associated with the largest value or the one with the smallest value of the parameter of interest, several procedures in the literature have been reviewed by Panchapakesan (1995) who has also considered variations and modifications of the basic selection goals. For detailed discussions of various aspects of the theory of selection and ranking and related problems, the reader is referred to Gibbons et al. (1977), Gupta and Panchapakesan (1979), and Bechhofer et al. (1995). Now, one can consider the goal of simultaneously selecting the extreme populations, namely, the populations associated with the largest and the smallest values of the parameter of interest. Mishra (1986) considered this goal under the IZ formulation for selecting fi'om a set of populations belonging to a one-parameter family where the parameter of interest is of location or scale. Mishra and Dudewicz (1987) have considered this goal under the SS formulation for selecting from normal populations in terms of means. Dhariyal and Misra (1994) have considered a Bayesian approach to this problem while Misra and Dhariyal (1994) have considered Bayes-P* and minimax rules. In the present paper, we consider selecting the extreme populations from a set of k two-parameter exponential populations with densities f(x; #t, ai) given by (1.1) for i = 1 , . . ,k. Section 2 discusses selection in terms of the location parameter under the IZ approach whereas Section 3 deals with the problem under the SS approach. Selection with respect to the scale parameter is addressed in Sections 4 and 5 under the IZ and SS approaches, respectively. The procedures discussed in Sections 2-5 are based on complete sample observations. Section 6 considers all the above selection problems based on Type-II censored samples in the context of life-testing experiments. Some remarks regarding future investigations are made in Section 6.

Simultaneous selection of extreme populations

815

2. Selection in terms of the loeation parameter; IZ approaeh Our goal is to select the two populations that are associated with the largest and the smallest #~, both identified correctly. We assume that al . . . . . «k = a. A CS occurs when a rule selects the two extreme populations correctly identified. Any valid rule R is required to satisfy P(CSIR ) _> P* whenever #[21 - #[1] ~

fT

and #N - il[k-l]

~ f~ '

(2.1)

where the positive constants f~ and f~ and the probability level P* are specified in advance by the experimenter. F o r a meaningful problem, we take (1/k(k - 1)) < P* < 1. The set O1 = {_~= ( # 1 , . . , # k ) : #[2]--#[11--> fiT, #[k]--#[k 1] ~ Ô~} is the PZ. We consider below two cases: a known and a unknown.

2.1. Known a case Let X~-I,...,Xi, be a r a n d o m sample of size n from Hi, i = 1 , . . , k . De#ne Y/= minl<j 2 and 0 < u < 1. Thus h(u) is concave on (0, 1). Further, limù_~0h(u)< 0 and lim~+l h(u)= 0. Therefore, there exists a unique solution for Eq. (2.6) for u C (0, 1). This leads to the determination of the minimum sample size n. Table 1 gives values of A = A(k,P*) for k = 2 ( 1 ) 1 0 , 15, 20, 25, 30 and P*= 0.90, 0.95 and 0.99.

2.2. Unknown « case When Œis known, the determination of the minimum sample size n required for the procedure R1 depends on the knowledge of «. When a is not known, we cannot determine the minimum sample size needed to satisfy the probability requirement (2.1). In fact, there does not exist a single sample procedure that can satisfy (2.1). We propose a two-stage procedure R2 described below. R2: (1) First, take n independent observations X/l,... ,X/n from ~Ii, i = 1 , . . , k, where n is arbitrarily chosen. Let

1 ä - k (1n ~

k ~'~ ~ ( X ~ j - Yi), i=1 j=l

where Yi = minl_<j y, a is the unique solution of

(2.7)

x f ( s ) i ( t ) 9~ (w)d« dt dw = P* ,

v = 2k(n 1) and 9u(-) is the density function of a chi-square variable with v degrees of freedom. (3) N o w take a second sample of size N - n from each population. Let Y~be the smallest of the N observations from IIi, i = 1 , . . , k, and let Y[1] _< "'" _< Y[kl denote the ordered Y/. Select the populations that yield Y3] and Y[k] as the populations associated with #[1] and #[k], respectively.

2.2.1. P C S and its infimum Letting, as before, Y(i) to be the statistic from the sample corresponding to the

population a s s o c i a t e d

with

#[i], i ---- 1 , . . ,

k, w e h a v e

P(CSIR2) = Pr[Y0)-< Y(i) -< Y(~),i = 2 , . . , k -

11

(3Q

= ZPr[N

= m]PrIY(1) < Y(i) Z

-rr, Pr[N = m]Pr [U1 - m~)*l ~r 2 and 0 < u < 1. Thus h(u) is concave on (0, 1). Further, l i m ù ~ 0 h ( u ) < 0, and l i m ù ~ l h ( u ) = 0. Therefore, there exists a unique solution for Eq. (3.6) for u E (0, 1). This yields the constant c by (3.5). Table 2 gives values o f c = c(k,P*) for k = 2(1)10, 15, 20, 25, 30, and P* = 0.90, 0.95 and 0.99.

3.1.2. Expected sizes of the selected subsets As p e r f o r m a n c e characteristics of procedure R3, we need to evaluate the suprema of E(B) and E(G), the expected sizes of the selected subsets B and G, respectively. Table 2 Value of c for which P(CSIR3 ) = P*

k•*

0.90

0.95

0.99

2 3 4 5 6 7 8 9 10 15 20 25 30

1.60943 2.43129 2.82533 3.09206 3.29626 3.46228 3.60395 3.72666 3.83533 4.24763 4.53695 4.76053 4.94288

2.30258 3.13741 3.53729 3.80739 4.01383 4.18199 4.32428 4.44791 4.55742 4.97203 5.26256 5.48681 5.66967

3.91203 4.75681 5.16116 5.43390 5.64193 5.81147 5.95455 6.07876 6.18894 6.60543 6.89681 7.12094 7.30525


821

We first consider E(G). Let Qi denote the probability that the p o p u l a t i o n associated with #[i] is included in the selected subset G. Then it is easily seen that E(G) = ~~=1 Qi, where _

-

-

C1(7]

Qi = Pr [Y(i) > Y[k] ~ - j =Pr

[YÜ) 3. It can be shown to be true for k = 2. In this case, we can write E(G) in the f o r m

E(G) =

/max{0,~-cl ~ }

[1 - e-(t-~+c~)]e -t dt +

= {2-1e-C'[e~+e 1 +

-~1

½e-~[e C' - e «~]

/0 ~

[1 - e-(«+«+Cl)]e -t dt

ifc~

b2 W«] vä , where W = - vn j a ~~]

b2wu 9v(w) dw , vn j

where gv(.) is the density of a chi-square variable with v degrees of freedom. Based on the arguments in the k n o w n a case,

[

blwa vn

Pr 17(1) -< Y [ [ 1 ] - + - - - , Y(k) >-- YN

-->[~[(t+l/v,)1m0in(b"b2)W[jo

b2wa] vn j

[_F(t-7~) -F(s -- ~W-)] k-2

× f ( s ) f ( t ) ds dt . Thus, in order to satisfy (3.8), the constants bl and b2 should be chosen to satisfy

fO°°~O°°~ot+(1/v)min(bl'b2)W[F(t@~) _F(s bS_)]k-2 × f ( s ) f ( t ) Ov(w)ds dt dw = P*

(3.9)

W h e n bi = b2 = b, Eq. (3.9) becomes

× f(«) f ( t ) gv (w) d« dt dw = P* .

(3.10)

A table of values of b satisfying (3.10) for selected values of k, n and P* is not available at present.

4. Selection in terms of the scale parameter; IZ approach O u r goal now is to select the two populations that are asso¢iated with the largest and the smallest ui, both identified corre¢tly. Let 0 < «[11 -~ "'" -~ aN denote the

823


ordered a» It is assumed that there is no prior information about the correct pairing of the ordered and the unordered O-i. A CS occurs when a rule selects the extreme populations correctly identified. Let

B2 = ~'O-= (O-1,.. ,O'k) : 0"[2] > ~~ > 1, O-[h] _> 61 > 1 ; , O-[1] O-[k-Il J t where 6~ and 6~ are specified in advance. Any valid rule R is required to satisfy: P(CSIR ) _> P*

whenever a E ~ 2 ,

(4.1)

where 1/k(k- 1) < P* < 1. We assume that the Pi are all known or all unknown. Let X~j, j = 1 , . . , n, be independent observations from Ili, i = 1 , . . , k. Define n

Si =

v

Pz)

[1i~/~~-~/

if

#i is

known, (4.2)

~~~,i~ ~nknown,

where Y,- = minl<j P*

for all a E ~2 ,

(5.1)

where ~2 = { a : _a = ( a l , . . . ,ffk), a > O,i = 1 , . . ,k} a n d

1 / k ( k - 1) < P* < 1.

Table 3 Value of v for which P(CSIR5) = 0.90

k•

2

2.5

3

3.5

4

2 3 4 5 6 7 8 9 10

3 12 16 18 20 21 22 23 23

2 7 9 11 12 13 13 14 14

1 5 7 8 8 9 9 10 10

1 4 5 6 7 7 8 9 9

1 3 4 5 5 6 6 6 7

Simultaneousselectionof extremepopulations

825

5.1. Procedure R6 Based on sample of size n from each population, we define Si as in (4.2). As before, S[1] _< .-. _< S N are the ordered Si. We now propose procedure R6:

Put IIi in SG iffSi > clS N and,

(5.2)

put IIi in SB iff Si _clS N, S ( 1 ) < I s [ l l ]

Il

S(1)~< min(ca

sk

c2) ( )1

= Pr[«2«Ill Tl _< ~ _ 2 because of estimating a.


828

6.3. Selection in terms of the scale parameter: I Z approach We assume that the #i are all k n o w n or all unknown, not necessarily equal. F o r the T y p e - I I censored sample f r o m IIi (1 < i < k), define ~] - Xi[i]) + (n - r) (Xi[r] - X/[1]

if #i is unknown,

It is k n o w n that T~ = v6i/«[i], i = 1 , . . ,k, are i.i.d, having a chi-square distribution with v degrees of freedom, where v = 2r when the #i are k n o w n and v = 2(r - 1) when the #i are u n k n o w n . Also, the di are independent of the ~[1]. The P Z is (22 defined in Section 4 for the complete sample case. Our procedure R~ is same as R» defined in (4.3) with S i replaced by di. Let ~[I] ~--- "'" ~ Ô-[k] denote the ordered ~[i]. Then the p r o p o s e d procedure is R~:

Select the populations that yield ä[1] and ô-N as those associated with «[1] and «[k], respectively .

The derivation of the PCS and its infimum over ~2 are the same as for procedure Rs. The m i n i m u m value of v (therefore, of r) is determined such that the righthand side of (4.4) [(4.5) when 6~ = 6~ = 6*] is at least P*. We note that we can have any n _> r to start with. In the case of u n k n o w n #i, we need r > 2.

6.4. Selection in terms of the scale parameter." SS approach In this case, out procedure R~ is same as R6 defined in (5.2) with Si replaced by äi. In other words, we p r o p o s e R~:

Put IIi in SG iff äi >_ e~ä[k] and 1 ^ put IIi in SB iff äi _< ~«[1] ,

where Cl and e2 are constants in the interval (0,1) to be chosen so that the probability requirement (5.1) is satisfied. With derivations parallel to the case of R 6 , the infimum of P(CSIR~) is given by (5.3) with v equal 2r or 2 ( r - 1) depending on whether the #i are all k n o w n or u n k n o w n , respectively. W h e n cl = c2 = c, then the values o f c for selected values o f k and v, and P* -- 0.90 are given in Table 4. We again note that the constant c is the same for any n > r and we need r > 2 when the #i are unknown.


829

7. Coneluding remarks I n o u r discussion o f several selection p r o c e d u r e s in the preceding sections, we have n o t c o n s i d e r e d some aspects o f these procedures. F o r example, SS p r o c e d u r e R3 involves two c o n s t a n t s Cl a n d c » W h e n we t a k e cl = c2 = c, the s o l u t i o n is unique. H o w e v e r , w h e n cl a n d c2 are n o t necessarily equal, there could be m a n y pairs o f (Cl, c2) satisfying the p r o b a b i l i t y requirement; in which case, one has to investigate the o p t i m a l s o l u t i o n b y considering, for example, E ( G ) + E ( B ) . These c o m m e n t s also a p p l y to other SS p r o c e d u r e s discussed earlier. In o u r f o r m u l a t i o n o f SS, o u r goal is to select two n o n - e m p t y subsets, B c o n t a i n i n g the lower extreme p o p u l a t i o n a n d G c o n t a i n i n g the other. It is n o t r e q u i r e d t h a t these two subsets should be n o n - o v e r l a p p i n g . The p r o b l e m to be investigated is p a r t i t i o n i n g o f the k p o p u l a t i o n s into three subsets B, M, G so t h a t B a n d G include the lower a n d the u p p e r extreme p o p u l a t i o n s , respectively, a n d M, the m i d d l e subset, c o u l d p o s s i b l y be empty. Some efforts have been m a d e in this direction b y the a u t h o r s b u t w i t h o u t m u c h success at this time. O n e can consider progressive T y p e - I I censored s a m p l i n g which is a generalization o f the s t a n d a r d T y p e - I I censoring c o n s i d e r e d here; see A g g a r w a l a a n d B a l a k r i s h n a n (1998) for some p r o p e r t i e s o f progressively c e n s o r e d - o r d e r statistics a n d related references. W e do n o t see a n y technical difficulties in extending o u r results u n d e r progressive censoring.

References Aggarwala, R. and N. Balakrishnan (1998). Some properties of progressive censored order statistics from arbitrary and uniform distributions with applications to inference and simulation. J. Stat. Plan. Inf 70, 35 49. Bechhofer, R. E. (1954). A single-sample multiple decision procedure for ranking means of normal populations with known variances. Arm. Math. Stat. 25, 16 39. Bechhofer, R. E., T. J. Santner and D. M. Goldsman (1995). Design and Analysis of Experimentsfor Statistical Selection, Screening and Multiple Comparisons. Wiley, New York. Dhariyal, I. D. and N. Misra (1994). Simultaneous selection of extreme populations: a Bayesian approach. Comm. Stat. Theor. Meth. 23, 1993~027. Gibbons, J. D., I. Olkin and M. Sobel (t977). Selecting and Ordering Populations: A New Statistical Methodology. Wiley, New York; Reprint: SIAM, Phlladelphia, 1999. Gupta, S. S. (1956). On a decision rule for a problem in ranking means. Ph.D. Dissertation, Institute of Statistics, University of North Carolina, Chapel Hi11, North Carolina. Gupta, S. S. (1963). On a selection and ranking procedure for gamma populations. Ann. Inst. Statist. Math. 14, 199-216. Gupta, S. S. and S. Panchapakesan (1979). Multiple Deeision Proeedures: Theory and Methodology of Selecting and Ranking Populations. Wiley, New York. Mishra, S. N. (1986). Simultaneous selection of extreme population means: indifference zone formulation. Am. J. Math. Management Sci. 6, 131 142. Mishra, S. N. and E. J. Dudewicz (1987). Simultaneous selection of extreme populations: A subset selection approach. Biomet. J. 4, 471-483. Misra, N. and I. D. Dhariyal (1994). Simultaneous selection of extreme populations: Bayes-P* and minimax rules. Comm. Star. - Theor. Meth. 23, 1963-1992.

830


Ofosu, J. B. (1972). On selection procedures for exponential distribution. Bull. Math. Stat. 16, 1-9. Panchapakesan, S. (1995). Selection and ranking procedures. In The Exponential Distribution." Theory Methods and Applications, pp. 259378 (Eds. N. Balakrishnan and A. P. Basu). Gordon and Breach, New York, New Jersey.

Subject Index

A consecutive k-out-of-n: F(G) 239 A priori distribution 487 Accelerated life test 623 lifetime tests 623 test 611 tests 458 Acceleration factor 626 function 625 Acceptance constant 463 number 458 sampling plans 415 Action space 486 Active redundancy 302 Adaptive rejection sampling 532 Aging 69 properties 307, 361 Alternating renewal process 806 Approximate estimators 431 Approximations 394 Arbitrary continuons distributions 386 Arrhenius function 612 law 626 Asymptotic covariance matrix 462 Attribute sampling 458 Audit laboratory testing 693 Availability 2, 7 Bartlett's test 322 Bathtub failure rate functions Bathtub-shaped 69 Bayes 529 decision function 487 risk 487 Bayesian 523, 527 530, 538 analysis 634, 733 models 708, 724 robustness 496

Best linear invariant estimation 400 Best linear unbiased estimates 432 estimation 336, 396 estimators 465 Beta 140 process 785 BFR 73 Blas 431 Binary multistate system 7 Bivariate normal distribution 641 Weibull distribution 652 Blackwell renewal theorem 17 BLUE 336, 344 Bounded relative error property 35 Burn-in 70, 185, 573 Burr 307 Cause-of-failure 529 Cause-specific 534 hazards 520 probability density function 520 Censored data 685 Weibull data 518 Censoring 669 Central limit theorem 18 Change point 74 Chapman-Kolmogorov equation 3 Characteristic Iife 612 Characterization 105, 106, 117, 121 Circular connected X-out-of-(n, m): F(G) lattice 240 Circular consecutive k-out-of-n: F systems 250 Closure properties 115, 124, 135 Coherent system 216 Common location parameter 351 Competing failure modes 604

189

831

832 risk 77 risk model 005 risks 523, 535 Component-level tests 659 Compound Gompertz 140 Weibull 140 Computer algebra systems 283 Conditional probability generating functions 282 Conditioning methods 28 Confidence intervals 790, 804 limits 616 Connected (r,s)-or-(s, r)-out-of-(n, m): F(6) 241 lattice system 241 Consecutive k-out-of-n systems 238 k-out-of-n: F 238 system 281 Consistency 781 Constant-stress modei 612 Construction techniques 81 Consumer's risk 460 Continuous time Markov chain 2 Correct selection (CS) 813 Correlation methods 26 Countable mixture 144 Counting process 43 Cox model 626 proportional hazard model 729 Cumulative exposure 614 exposure model 623, 627 TTT-statistic 476 Cutset 292 Decision procedures 485 Degradation data 458 Degrees of censoring 460 Demonstration testing 682 Dependent competing risks 519 Detection ofwearont 807 Deterioration 44 DFR 361 Diagnostic probability 500 Directed Markov distribution 284 tree 283 Dirichlet process 785

Subject index Discrete bathtub shape 95 distribution theory 282 time Markov chains 2 Z2 distribution 791 Distributions of stopping times 789, 792, 798 Dominance relations 105-107, 114, 117 Double proofload design 646 Doubly truncated exponential distribution 389 Duane model 715 Dynamic importance sampling measure 33 stresses 611 Embedded Markov chain 15 method 38 Equilibrium distribution in higher dimension 106, 124, 135 Ergodic Markov chain 6 Expected Fisher information 438 Expected stopping time 800, 803 times 789 ExponentiaI 630 distribution 303, 377, 460, 783, 789, 792 distributions 307, 330 power 78 Exponentiated Weibull 79 Extreme-value distribution 462 distributions 383 Eyring law 627 Eyring-Weibull model 613 Failure intensity function 709, 711,722 mode 538 modes 523 process 708 rate 69 rate ordering 365 truncated 49 Failure-step stress test 624 Field life 611 reliability 585 Finite mixture 144 Finite-state Markov chain 245 Fisher information 438,642 matrix 463, 710 Follow-up survey 587

Subject index Freund's bivariate exponential distribution 328 Full conditional 531 Functional central limit theorem

19

Gamma distribution 196, 319 mixture 82 Gauss hypergeometric distribution 359 Generalized beta distribution 356, 358 Eyring relationship 612 order statistics 311, 314 Pareto distribution 307 Generating function 296 functions 267, 799 Gibbs sampler 530 Goel-Okumoto model 711 Gompertz 140 Goodness of fit 538 tests 57 Graphical interpretation 716, 717 Hartley's test 323 Hazard function 200, 207, 310, 777, 781 Hazard measure 199201, 209 211 order 203 ordering 199, 200, 203206 Hazard rate 199, 208210, 212 function 142 functions 189 order 202 ordering 199, 203,210 Heterogeneous populations 140, 185 Hierarchical structures 227 Higher order equilibrium distribution 111, 125, 132 Hm ordering 206, 208 Homogeneous Markov chain 3 populations 140 Hr ordering 205 Hypergeometric function 356 Identifiability 146 Identification probability 499 IFR 361 IFRA 362 Importance sampling 2, 29, 32 Indicator of the event 266 Indifference zone 813 Inference 373

833

Intensity function 43 Inter-event times 474 Interval censored 538 censoring 533 Inverse Gaussian-Weibull mixed model (IG-W) 168 power function 612 Power Law 626 transform method 23 Inverted gamma distribution 318 Jeffrey's prior 528, 530 Jelinski 721 )th component in the mixture

I44

k-out-of-n system 215 systems 302 k-out-of-n: G systems 238 ( k - 1)-step Markov dependence kth record values 314

247

Laplac~Stieltjes transform 14 Laplace's trend test 478 Latent variables 530, 533 Law of large numbers 17 Least squares median ranks estimator 407 Lieberman and Resnikoff (1955) procedure 459 Life distribution 611 distributions 140 testing 139, 373 Lifetime 292 Likelihood function 779 Likelihood ratio 26 confidence limits 617 ordering 366 Linear connected X-out-of-(n, m): F(G) lattice system 240 Linear consecutive k-out-of-n: F systems 241 Linear-hazard-rate 196 Location-scale 530, 534 distribution 459 family 306, 326 Log-concave 533 Log-concavity 531 Logistic distribution 431 Lognormal distribution 620 Log-power model 717

834 Lomax distributions 307, 333 Loss of utility 486 Maintainability 7 Majorization 216 Marginal count data 595 Markov chain Monte Carlo (MCMC) 529, 733 Monte Carlo techniques 782 of first-order 249 Markov model 708, 721 Markov renewal equation 16 process 15 Markov trees 284 Masked 523, 524, 532, 535 cause-of-failure 524, 536, 538 group 499, 500 Masking 524, 529, 534 group g 499 probability 500 Mathematical programming 661 Maximum likelihood 651,709 fitting 615 estimation 333, 336, 395, 630 estimator 783, 784 estimators (MLEs) 306, 317, 431,460 (ML) estimates 681 MCMC 530-533 Mean cumulative function (MCF) 693, 694 down time 8 Mean residual life 70, 222 function 199, 200, 206 ordering 199, 200 Mean squared error 431 Mean time between failure 8 to failure 8 to repair 8 Mean up time 8 Mean value function 709 Measure Specific Dynamic Importance Sampling 34 Method of competing risk 36 Miner's rule 615, 620 Minimal cut 266 Minimal repair 46, 471 scheme 316 Minimum random subset 525 Mission time 92 Mixing proportion 144 Mixture 143 of distributions 186

Subject index Mixtures of exponential components 157 Gompertz components 162 Inverse Gaussian components 153 lognormal components 149 normal components 147 Rayleigh components 159 Weibull components 160 ML estimates 616 MLE 306, 336, 344, 354 Model of additive accumulation of damages 626 Modified semi-Poisson model 172 Modular decomposition 220 Modulated power law process 66 Moments 373 Monte Carlo methods 2 MRL function 222 order 202 ordering 205 MRS 533 Multiple step-stress test 624 systems 60 Musa-Okumoto model 714 New products 545 No-data problem 489 Node criticality 217 Non-homogeneous discrete time Markov chain 9 Poisson process 315, 316, 471, 708 Non-identifiability 530, 535 Non-monotonic aging classes 91 Nonparametric 653 Normal distribution 409 Norma~exponential mixed model 166 Observed Fisher information 438 OC curve 459~461 One-dimensional case 239 One-parameter models 396 One-sided sampling plans 420 Operating characteristic 800 characteristics 789 Optimal censoring schemes 373 test plans 630, 637 Optimization 663 Order statistic 303

Subject index statistics 200, 206,212, 295, 311,314, 750, 754, 772 Parallel system 13 systems 670 Parametric empirical Bayes models 65 Pareto distribution 140, 307, 333, 382 Partial masking 525 ordering 199, 200, 203, 307 Partially accelerated life test 624 Pearson distributions 332 I distributions 307 Perfect repair 46 Permutation equivalent 217 Pfeifer's records 311,314 Phase-type distributions 13 Piecewise exponential distributions 190 Pivotal quantities 431 quantity 690 Poincaré equations 228 Poisson binomiaI variables 216 process 43, 783, 790, 792 Population cdf 617 reliability 611 Posterior distribution 781,784 Power-law process 43, 475, 480 Preference zone 814 Prior distribution 782, 784 Probability coverages 431 generating functions 282 plot 750, 772 Producers risk 459 Progressive censoring 373 Type II censored order statistics 311 Type II censoring 314, 432 Type-I right censoring 432 Progressively Type-II censored samples 431 Proof load 641 Proportional closeness 790, 791,793, 803 failure rate 221 MRL 224 Proportional hazard model 195, 626 models 512 Pseudo-likelihood 599

Quality control

835 569

Rare event estimation 2 Reconstruction of reliability polynomial 233 Record valnes 200, 311,314 Recursive computation 388 Redundancy 568 Redundant cold standby system 13 systems 302 Regenerative simulation 33 Regrets 487 Relative error 29 Relayed bipolar consecutive k-out-of-n: F system 239 consecutive k-out-of-n: F 239 unipolar consecutive k-out-of-n: F system 239 Release time modeling 730 Relevation transform 316 Reliability 2, 7, 45, 139, 227, 281, 373,457 improvement 44, 568 testing 789, 802 tests 659 Reliability polynomial 227 of monotone structures 228 Renewal process 46 repair 46 Repair 570 Repairable system 131, 132, 790 systems 45, 105, 698, 699, 805 Residual lifetime 69 Reverse hazard measure 210 rate 210 Reverse time hazard function 594 Roller-coaster 99 Sample size 463 determination 679 Sampling acceptance 789 Scale mixtures 196 Scaled TTT-transform 470, 472, 473 Schick 723 Sectional models 74, 77 Selection and ranking 813 Semi-Markov model 2 processes 2 Sequential

836

Subject index

k-out-of-n systems 303,304 estimation 790, 795, 803 probability ratio test 797 Sequential-order statistics 303, 309, 311 Series system 13 systems 666 Shape parameter 612 Shock models 315 Simple step-stress test 624 Simulation 373, 682 Simultaneous selection of extreme populations 813 Software reliability 707, 733, 808 growth model 709, 720 models 707 Specification limits 459 S-shaped models 713 St ordering 206 Stabile stress distribution 619 State of nature 485 Stationary Markov chain 246 Step-stress 611,623 testing 623 Stochastic order 202 ordering 200, 204, 365 Stopping variable 791 Stratified sampling 28 Stress corrosion crack initiation 680 function 625 profiles 611,614 Stres~strength models 354 Strong Markov property 792 Subset selection (SS) 813 Sums ofindependent random variable 200,206 Supermajorized 216 Survival analysis 789 Symmetric proof load design 649 Symmetry assumption 501,526 System reliability 693, 694, 704 System-based component test plans 659 System-level tests 659 Taboo probability 245 Tail-ordering 366 Taylor approximation 463 Test plans 621 Tests of exponentiality 83 Time stress pattern 624 truncated 49

Time-step stress test 624 Time-varying stresses 611 TTT-plot 85, 469, 472 Two-dimensional consecutive k-out-of-n: F systems 240, 268 Two-parameter exponential distribution 462 models 401 Two-sided sampling plans 424 Two-stage optimization 662 Type I censoring 432, 457, 624 Type-II censoring 432, 458,463,024 progressive censoring 458 UMVUE 336, 344, 354 Uniform distribution 307, 379 Uniformly minimum variance unbiased estimation 336 estimators (UMVUEs) 306 Upside-down bathtub 88 Used products 549 U-shape 70 Variables-sampling plans 458,461 Variance reduction methods 2, 25 Warrant policies combination 547 cumulative 548 extended 551 FRW 546 PRW 547 RIW 549 Warranty cost basis 556 costs 551, 593 data 577, 585 reserves 575 servicing 574 Weak submajorization 216 Wearout 72, 807 Weibull 140 distribution 195, 307, 336, 383, 462, 600, 612, 681,715, 749, 750, 753, 772, 781 distributions 331 lifetime 631 mixture 82 nonhomogeneous Poisson process 44 process 43, 715 Weibull-exponential mixed model 172 Weight function 213, 214

Subject index Weighted least squares 431 estimators 433 Weighted random variable 213, 214

Weinman multivariate exponentiaI distribution 328 "Worth" of the experiment or test 487

837

Handbook of Statistics Contents of Previous Volumes

Volume 1. Analysis of Variance Edited by P. R. Krishnaiah 1980 xviii + 1002 pp.

1. Estimation of Variance Components by C. R. Rao and J. Kleffe 2. Multivariate Analysis of Variance of Repeated Measurements by N. H. Timm 3. Growth Curve Analysis by S. Geisser 4. Bayesian Inference in MANOVA by S. J. Press 5. Graphical Methods for Internal Comparisons in ANOVA and MANOVA by R. Gnanadesikan 6. Monotonicity and Unbiasedness Properties of ANOVA and MANOVA Tests by S. Das Gupta 7. Robustness of ANOVA and MANOVA Test Procedures by P. K. Ito 8. Analysis of Variance and Problem under Time Series Models by D. R. Brillinger 9. Tests of Univariate and Multivariate Normality by K. V. Mardia 10. Transformations to Normality by G. Kaskey, B. Kolman, P. R. Krishnaiah and L. Steinberg 11. ANOVA and MANOVA: Models for Categorical Data by V. P. Bhapkar 12. Inference and the Structural Model for ANOVA and MANOVA by D. A. S. Fraser 13. Inference Based on Conditionally Specified ANOVA Models Incorporating Preliminary Testing by T. A. Bancroft and C. -P. Han 14. Quadratic Forms in Normal Variables by C. G. Khatri 15. Generalized Inverse of Matrices and Applications to Linear Models by S. K. Mitra 16. Likelihood Ratio Tests for Mean Vectors and Covariance Matrices by P. R. Krishnaiah and J. C. Lee

839

840 17. 18. 19. 20. 21. 22. 23. 24. 25.

Contents of previous volumes

Assessing Dimensionality in Multivariate Regression by A. J. Izenman Parameter Estimation in Nonlinear Regression Models by H. Bunke Early History of Multiple Comparison Tests by H. L. Harter Representations of Simultaneous Pairwise Comparisons by A. R. Sampson Simultaneous Test Procedures for Mean Vectors and Covariance Matrices by P. R. Krishnaiah, G. S. Mudholkar and P. Subbiah Nonparametric Simultaneous Inference for Some MANOVA Models by P. K. Sen Comparison of Some Computer Programs for Univariate and Multivariate Analysis of Variance by R. D. Bock and D. Brandt Computations of Some Multivariate Distributions by P. R. Krishnaiah Inference on the Structure of Interaction in Two-Way Classification Model by P. R. Krishnaiah and M. Yochmowitz

Volume 2. Classification, Pattern Recognition and Reduction of Dimensionality Edited by P. R. Krishnaiah and L. N. Kanal 1982 xxii + 903 pp.

1. Discriminant Analysis for Time Series by R. H. Shumway 2. Optimum Rules for Classification into Two Multivariate Normal Populations with the Same Covariance Matrix by S. Das Gupta 3. Large Sample Approximations and Asymptotic Expansions of Classification Statistics by M. Siotani 4. Bayesian Discrimination by S. Geisser 5. Classification of Growth Curves by J. C. Lee 6. Nonparametric Classification by J. D. Broffitt 7. Logistic Discrimination by J. A. Anderson 8. Nearest Neighbor Methods in Discrimination by L. Devroye and T. J. Wagner 9. The Classification and Mixture Maximum Likelihood Approaches to Cluster Analysis by G. J. McLachlan 10. Graphical Techniques for Multivariate Data and for Clustering by J. M. Chambers and B. Kleiner 11. Cluster Analysis Software by R. K. Blashfield, M. S. Aldenderfer and L. C. Morey 12. Single-link Clustering Algorithms by F. J. Rohlf 13. Theory of Multidimensional Scaling by J. de Leeuw and W. Heiser 14. Multidimensional Scaling and its Application by M. Wish and J. D. Carroll 15. Intrinsic Dimensionality Extraction by K. Fukunaga

Contents of previous voßmes

841

16. Structural Methods in Image Analysis and Recognition by L. N. Kanal, B. A. Lambird and D. Lavine 17. Image Models by N. Ahuja and A. Rosenfeld 18. Image Texture Survey by R. M. Haralick 19. Applications of Stochastic Languages by K. S. Fu 20. A Unifying Viewpoint on Pattern Recognition by J. C. Simon, E. Backer and J. Sallentin 21. Logical Functions in the Problems of Empirical Prediction by G. S. Lbov 22. Inference and Data Tables and Missing Values by N. G. Zagoruiko and V. N. Yolkina 23. Recognition of Electrocardiographic Patterns by J. H. van Bemmel 24. Waveform Parsing Systems by G. C. Stockman 25. Continuous Speech Recognition: Statistical Methods by F. Jelinek, R. L. Mercer and L. R. Bahl 26. Applications of Pattern Recognition in Radar by A. A. Grometstein and W. H. Schoendorf 27. White Blood Cell Recognition by E. S. Gelsema and G. H. Landweerd 28. Pattern Recognition Techniques for Remote Sensing Applications by P. H. Swain 29. Optical Character Recognition Theory and Practice by G. Nagy 30. Computer and Statistical Considerations for Oil Spill Identification by Y. T. Chinen and T. J. Killeen 31. Pattern Recognition in Chemistry by B. R. Kowalski and S. Wold 32. Covariance Matrix Representation and Object-Predicate Symmetry by T. Kaminuma, S. Tomita and S. Watanabe 33. Multivariate Morphometrics by R. A. Reyment 34. Multivariate Analysis with Latent Variables by P. M. Bentler and D. G. Weeks 35. Use of Distance Measures, Information Measures and Error Bounds in Feature Evaluation by M. Ben-Bassat 36. Topics in Measurement Selection by J. M. Van Campenhout 37. Selection of Variables Under Univariate Regression Models by P. R. Krishnaiah 38. On the Selection of Variables Under Regression Models Using Krishnaiah's Finite Intersection Tests by J. L Schmidhammer 39. Dimensionality and Sample Size Considerations in Pattern Recognition Practice by A. K. Jain and B. Chandrasekaran 40. Selecting Variables in Discriminant Analysis for Improving upon Classical Procedures by W. Schaafsma 41. Selection of Variables in Discriminant Analysis by P. R. Krishnaiah

842


Volume 3. Time Series in the Frequency D o m a i n Edited by D. R. Brillinger and P. R. Krishnaiah 1983 xiv + 485 pp.

1. Wiener Filtering (with emphasis on frequency-domain approaches) by R. J. Bhansali and D. Karavellas 2. The Finite Fourier Transform of a Stationary Process by D. R. Brillinger 3. Seasonal and Calender Adjustment by W. S. Cleveland 4. Optimal Inference in the Frequency Domain by R. B. Davies 5. Applications of Spectral Analysis in Econometrics by C. W. J. Granger and R. Engle 6. Signal Estimation by E. J. Hannan 7. Complex Demodulation: Some Theory and Applications by T. Hasan 8. Estimating the Gain of a Linear Filter from Noisy Data by M. J. Hinich 9. A Spectral Analysis Primer by L. H. Koopmans 10. Robust-Resistant Spectral Analysis by R. D. Martin 11. Autoregressive Spectral Estimation by E. Parzen 12. Threshold Autoregression and Some Frequency-Domain Characteristics by J. Pemberton and H. Tong 13. The Frequency-Domain Approach to the Analysis of Closed-Loop Systems by M. B. Priestley 14. The Bispectral Analysis of Nonlinear Stationary Time Series with Reference to Bilinear Time-Series Models by T. Subba Rao 15. Frequency-Domain Analysis of Multidimensional Time-Series Data by E. A. Robinson 16. Review of Various Approaches to Power Spectrum Estimation by P. M. Robinson 17. Cumulants and Cumulant Spectral Spectra by M. Rosenblatt 18. Replicated Time-Series Regression: An Approach to Signal Estimation and Detection by R. H. Shumway 19. Computer Programming of Spectrum Estimation by T. Thrall 20. Likelihood Ratio Tests on Covariance Matrices and Mean Vectors of Complex Multivariate Normal Populations and their Applications in Time Series by P. R. Krishnaiah, J. C. Lee and T. C. Chang


843

Volume 4. Nonparametric Methods Edited by P. R. Krishnaiah and P. K. Sen 1984 xx + 968 pp.

1. Randomization Procedures by C. B. Bell and P. K. Sen 2. Univariate and Multivariate Mutisample Location and Scale Tests by V. P. Bhapkar 3. Hypothesis of Symmetry by M. Hugkovä 4. Measures of Dependence by K. Joag-Dev 5. Tests of Randomness against Trend or Serial Correlations by G. K. Bhattacharyya 6. Combination of Independent Tests by J. L. Folks 7. Combinatorics by L. Takäcs 8. Rank Statistics and Limit Theorems by M. Ghosh 9. Asymptotic Comparison of Tests A Review by K. Singh 10. Nonparametric Methods in Two-Way Layouts by D. Quade 11. Rank Tests in Linear Models by J. N. Adichie 12. On the Use of Rank Tests and Estimates in the Linear Model by J. C. Aubuchon and T. P. Hettmansperger 13. Nonparametric Preliminary Test Inference by A. K. Md. E. Saleh and P. K. Sen 14. Paired Comparisons: Some Basic Procedures and Examples by R. A. Bradley 15. Restricted Alternatives by S. K. Chatterjee 16. Adaptive Methods by M. Hugkovä 17. Order Statistics by J. Galambos 18. Induced Order Statistics: Theory and Applications by P. K. Bhattacharya 19. Empirical Distribution Function by E. Csäki 20. Invariance Principles for Empirical Processes by M. Csörgö 21. M-, L- and R-estimators by J. Jureökovä 22. Nonparametric Sequantial Estimation by P. K. Sen 23. Stochastic Approximation by V. Dupa6 24. Density Estimation by P. Révész 25. Censored Data by A. P. Basu 26. Tests for Exponentiality by K. A. Doksum and B. S. Yandell 27. Nonparametric Concepts and Methods in Reliability by M. Hollander and F. Proschan 28. Sequential Nonparametric Tests by U. Müller-Funk 29. Nonparametric Procedures for some Miscellaneous Problems by P. K. Sen 30. Minimum Distance Procedures by R. Beran 31. Nonparametric Methods in Directional Data Analysis by S. R. Jammalamadaka 32. Application of Nonparametric Statistics to Cancer Data by H. S. Wieand

844


33. Nonparametric Frequentist Proposals for Monitoring Comparative Survival Studies by M. Gail 34. Meterological Applications of Permutation Techniques based on Distance Functions by P. W. Mielke, Jr. 35. Categorical Data Problems Using Information Theoretic Approach by S. Kullback and J. C. Keegel 36. Tables for Order Statistics by P. R. Krishnaiah and P. K. Sen 37. Selected Tables for Nonparametric Statistics by P. K. Sen and P. R. Krishnaiah

Volume 5. Time Series in the Time D o m a i n Edited by E. J. Hannan, P. R. Krishnaiah and M. M. R a o 1985 xiv + 490 pp.

1. Nonstationary Autoregressive Time Series by W. A. Fuller 2. Non-Linear Time Series Models and Dynamical Systems by T. Ozaki 3. Autoregressive Moving Average Models, Intervention Problems and Outlier Detection in Time Series by G. C. Tiao 4. Robustness in Time Series and Estimating ARMA Models by R. D. Martin and V. J. Yohai 5. Time Series Analysis with Unequally Spaced Data by R. H. Jones 6. Various Model Selection Techniques in Time Series Analysis by R. Shibata 7. Estimation of Parameters in Dynamical Systems by L. Ljung 8. Recursive Identification, Estimation and Control by P. Young 9. General Structure and Parametrization of ARMA and State-Space Systems and its Relation to Statistical Problems by M. Deistler 10. Harmonizable, Cramér, and Karhunen Classes of Processes by M. M. Rao 11. On Non-Stationary Time Series by C. S. K. Bhagavan 12. Harmonizable Filtering and Sampling of Time Series by D. K. Chang 13. Sampling Designs for Time Series by S. Cambanis 14. Measuring Attenuation by M. A. Cameron and P. J. Thomson 15. Speech Recognition Using LPC Distance Measures by P. J. Thomson and P. de Souza 16. Varying Coefficient Regression by D. F. Nicholls and A. R. Pagan 17. Small Samples and Large Equation Systems by H. Theil and D. G. Fiebig


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Volume 6. Sampling Edited by P. R. Krishnaiah and C. R. R a o 1988 xvi + 594 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24.

A Brief History of Random Sampling Methods by D. R. Bellhouse A First Course in Survey Sampling by T. Dalenius Optimality of Sampling Strategies by A. Chaudhuri Simple Random Sampling by P. K. Pathak On Single Stage Unequal Probability Sampling by V. P. Godambe and M. E. Thompson Systematic Sampling by D. R. Bellhouse Systematic Sampling with Illustrative Examples by M. N. Murthy and T. J. Rao Sampling in Time by D. A. Binder and M. A. Hidiroglou Bayesian Inference in Finite Populations by W. A. Ericson Inference Based on Data from Complex Sample Designs by G. Nathan Inference for Finite Population Quantiles by J. Sedransk and P. J. Smith Asymptotics in Finite Population Sampling by P. K. Sen The Technique of Replicated or Interpenetrating Samples by J. C. Koop On the Use of Models in Sampling from Finite Populations by I. Thomsen and D. Tesfu The Prediction Approach to Sampling theory by R. M. Royall Sample Survey Analysis: Analysis of Variance and Contingency Tables by D. H. Freeman, Jr. Variance Estimation in Sample Surveys by J. N. K. Rao Ratio and Regression Estimators by P. S. R. S. Rao Role and Use of Composite Sampling and Capture-Recapture Sampling in Ecological Studies by M. T. Boswell, K. P. Burnham and G. P. Patil Data-based Sampling and Model-based Estimation for Environmental Resources by G. P. Patil, G. J. Babu, R. c. Hennemuth, W. L. Meyers, M. B. Rajarshi and C. Taillie On Transect Sampling to Assess Wildlife Populations and Marine Resources by F. L. Ramsey, C. E. Gares, G. P. Patil and C. Taillie A Review of Current Survey Sampling Methods in Marketing Research (Telephone, Mall Intercept and Panel Surveys) by R. Velu and G. M. Naidu Observational Errors in Behavioural Traits of Man and their Implications for Genetics by P. V. Sukhatme Designs in Survey Sampling Avoiding Contiguous Units by A. S. Hedayat, C. R. Rao and J. Stufken

846


Volume 7. Quality Control and Reliability Edited by P. R. Krishnaiah and C. R. R a o 1988 xiv + 503 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Transformation of Western Style of Management by W. Edwards Deming Software Reliability by F. B. Bastani and C. V. Ramamoorthy Stress-Strength Models for Reliability by R. A. Johnson Approximate Computation of Power Generating System Reliability Indexes by M. Mazumdar Software Reliability Models by T. A. Mazzuchi and N. D. Singpurwalla Dependence Notions in Reliability Theory by N. R. Chaganty and K. Joag-dev Application of Goodness-of-Fit Tests in Reliability by H. W. Block and A. H. Moore Multivariate Nonparametric Classes in Reliability by H. W. Block and T. H. Savits Selection and Ranking Procedures in Reliability Models by S. S. Gupta and S. Panchapakesan The Impact of Reliability Theory on Some Branches of Mathematics and Statistics by P. J. Boland and F. Proschan Reliability Ideas and Applications in Economics and Social Sciences by M. C. Bhattacharjee Mean Residual Life: Theory and Applications by F. Guess and F. Proschan Life Distribution Models and Incomplete Data by R. E. Barlow and F. Proschan Piecewise Geometric Estimation of a Survival Function by G. M. Mimmack and F. Proschan Applications of Pattern Recognition in Failure Diagnosis and Quality Control by L. F. Pau Nonparametric Estimation of Density and Hazard Rate Functions when Samples are Censored by W. J. Padgett Multivariate Process Control by F. B. Alt and N. D. Smith QMP/USP-A Modern Approach to Statistical Quality Auditing by B. Hoadley Review About Estimation of Change Points by P. R. Krishnaiah and B. Q. Miao Nonparametric Methods for Changepoint Problems by M. Csögö and L. Horväth Optimal Allocation of Multistate Components by E. E1-Neweihi, F. Proschan and J. Sethuraman Weibull, Log-Weibull and Gamma Order Statistics by H. L. Harter Multivariate Exponential Distributions and their Applications in Reliability by A. P. Basu


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24. Recent Developments in the Inverse Gaussian Distribution by S. Iyengar and G. Patwardhan

Volume 8. Statistical Methods in Biological and Medical Sciences Edited by C. R. R a o and R. C h a k r a b o r t y 1991 xvi + 554 pp.

1. Methods for the Inheritance of Qualitative Traits by J. Rice, R. Neuman and S. O. Moldin 2. Ascertainment Biases and their Resolution in Biological Surveys by W. J. Ewens 3. Statistical Considerations in Applications of Path Analytical in Genetic Epidemiology by D. C. Rao 4. Statistical Methods for Linkage Analysis by G. M. Lathrop and J. M. Lalouel 5. Statistical Design and Analysis of Epidemiologic Studies: Some Directions of Current Research by N. Breslow 6. Robust Classification Procedures and Their Applications to Anthropometry by N. Balakrishnan and R. S. Ambagaspitiya 7. Analysis of Population Structure: A Comparative Analysis of Different Estimators of Wright's Fixation Indices by R. Chakraborty and H. DankerHopfe 8. Estimation of Relationships from Genetic Data by E. A. Thompson 9. Measurement of Genetic Variation for Evolutionary Studies by R. Chakraborty and C. R. Rao 10. Statistical Methods for Phylogenetic Tree Reconstruction by N. Saitou 11. Statistical Models for Sex-Ratio Evolution by S. Lessard 12. Stochastic Models of Carcinogenesis by S. H. Moolgavkar 13. An Application of Score Methodology: Confidence Intervals and Tests of Fit for One-Hit-Curves by J. J. Gart 14. Kidney-Survival Analysis of IgA Nephropathy Patients: A Case Study by O. J. W. F. Kardaun 15. Confidence Bands and the Relation with Decision Analysis: Theory by O. J. W. F. Kardaun 16. Sample Size Determination in Clinical Research by J. Bock and H. Toutenburg

848


Volume 9. Computational Statistics Edited by C. R. R a o 1993 xix + 1045 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Algorithms by B. Kalyanasundaram Steady State Analysis of Stochastic Systems by K. Kant Parallel Computer Architectures by R. Krishnamurti and B. Narahari Database Systems by S. Lanka and S. Pal Programming Languages and Systems by S. Purushothaman and J. Seaman Algorithms and Complexity for Markov Processes by R. Varadarajan Mathematical Programming: A Computational Perspective by W. W. Hager, R. Horst and P. M. Pardalos Integer Programming by P. M. Pardalos and Y. Li Numerical Aspects of Solving Linear Lease Squares Problems by J. L. Barlow The Total Least Squares Problem by S. Van Huffel and H. Zha Construction of Reliable Maximum-Likelihood-Algorithms with Applications to Logistic and Cox Regression by D. Böhning Nonparametric Function Estimation by T. Gasser, J. Engel and B. Seifert Computation Using the QR Decomposition by C. R. Goodall The EM Algorithm by N. Laird Analysis of Ordered Categorial Data through Appropriate Scaling by C. R. Rao and P. M. Caligiuri Statistical Applications of Artificial Intelligence by W. A. Gale, D. J. Hand and A. E. Kelly Some Aspects of Natural Language Processes by A. K. Joshi Gibbs Sampling by S. F. Arnold Bootstrap Methodology by G. J. Babu and C. R. Rao The Art of Computer Generation of Random Variables by M. T. Boswell, S. D. Gore, G. P. Patil and C. Taillie Jackkuife Variance Estimation and Bias Reduction by S. Das Peddada Designing Effective Statistical Graphs by D. A. Burn Graphical Methods for Linear Models by A. S. Hadi Graphics for Time Series Analysis by H. J. Newton Graphics as Visual Language by T. Selker and A. Appel Statistical Graphics and Visualization by E. J. Wegman and D. B. Carr Multivariate Statistical Visualization by F. W. Young, R. A. Faldowski and M. M. McFarlane Graphical Methods for Process Control by T. L. Ziemer


849

Volume 10. Signal Processing and its Applications Edited by N. K. Bose and C. R. R a o 1993 xvii + 992 pp.

1. Signal Processing for Linear Instrumental Systems with Noise: A General Theory with Illustrations for Optical Imaging and Light Scattering Problems by M. Bertero and E. R. Pike 2. Boundary Implication Rights in Parameter Space by N. K. Bose 3. Sampling of Bandlimited Signals: Fundamental Results and Some Extensions by J. L. Brown, Jr. 4. Localization of Sources in a Sector: Algorithms and Statistical Analysis by K. Buckley and X.-L. Xu 5. The Signal Subspace Direction-of-Arrival Algorithm by J. A. Cadzow 6. Digital Differentiators by S. C. Dutta Roy and B. Kumar 7. Orthogonal Decompositions of 2D Random Fields and their Applications for 2D Spectral Estimation by J. M. Francos 8. VLSI in Signal Processing by A. Ghouse 9. Constrained Beamforming and Adaptive Algorithms by L. C. Godara 10. Bispectral Speckle Interferometry to Reconstruct Extended Objects from Turbulence-Degraded Telescope Images by D. M. Goodman, T. W. Lawrence, E. M. Johansson and J. P. Fitch 11. Multi-Dimensional Signal Processing by K. Hirano and T. Nomura 12. On the Assessment of Visual Communication by F. O. Huck, C. L. Fales, R. Alter-Gartenberg and Z. Rahman 13. VLSI Implementations of Number Theoretic Concepts with Applications in Signal Processing by G. A. Jullien, N. M. Wigley and J. Reilly 14. Decision-level Neural Net Sensor Fusion by R. Y. Levine and T. S. Khuon 15. Statistical Algorithms for Noncausal Gauss Markov Fields by J. M. F. Moura and N. Balram 16. Subspace Methods for Directions-of-Arrival Estimation by A. Paulraj, B. Ottersten, R. Roy, A. Swindlehurst, G. Xu and T. Kailath 17. Closed Form Solution to the Estimates of Directions of Arrival Using Data from an Array of Sensors by C. R. Rao and B. Zhou 18. High-Resolution Direction Finding by S. V. Schell and W. A. Gardner 19. Multiscale Signal Processing Techniques: A Review by A. H. Tewfik, M. Kim and M. Deriche 20. Sampling Theorems and Wavelets by G. G. Walter 21. Image and Video Coding Research by J. W. Woods 22. Fast Algorithms for Structured Matrices in Signal Processing by A. E. Yagle

850


Volume 11. Econometrics Edited by G. S. Maddala, C. R. Rao and H. D. Vinod 1993 xx + 783 pp.

1. Estimation from Endogenously Stratified Samples by S. R. Cosslett 2. Semiparametric and Nonparametric Estimation of Quantal Response Models by J. L. Horowitz 3. The Selection Problem in Econometrics and Statistics by C. F. Manski 4. General Nonparametric Regression Estimation and Testing in Econometrics by A. Ullah and H. D. Vinod 5. Simultaneous Microeconometric Models with Censored or Qualitative Dependent Variables by R. Blundell and R. J. Smith 6. Multivariate Tobit Models in Econometrics by L.-F. Lee 7. Estimation of Limited Dependent Variable Models under Rational Expectations by G. S. Maddala 8. Nonlinear Time Series and Macroeconometrics by W. A. Brock and S. M. Potter 9. Estimation, Inference and Forecasting of Time Series Subject to Changes in Time by J. D. Hamilton 10. Structural Time Series Models by A. C. Harvey and N. Shephard 11. Bayesian Testing and Testing Bayesians by J. -P. Florens and M. Mouchart 12. Pseudo-Likelihood Methods by C. Gourieroux and A. Monfort 13. Rao's Score Test: Recent Asymptotic Results by R. Mukerjee 14. On the Strong Consistency of M-Estimates in Linear Models under a General Discrepancy Function by Z. D. Bai, Z. J. Liu and C. R. Rao 15. Some Aspects of Generalized Method of Moments Estimation by A. Hall 16. Efficient Estimation of Models with Conditional Moment Restrictions by W. K. Newey 17. Generalized Method of Moments: Econometric Applications by M. Ogaki 18. Testing for Heteroskedasticity by A. R. Pagan and Y. Pak 19. Simulation Estimation Methods for Limited Dependent Variable Models by V. A. Hajivassiliou 20. Simulation Estimation for Panel Data Models with Limited Dependent Variable by M. P. Keane 21. A Perspective on Application of Bootstrap methods in Econometrics by J. Jeong and G. S. Maddala 22. Stochastic Simulations for Inference in Nonlinear Errors-in-Variables Models by R. S. Mariano and B. W. Brown 23. Bootstrap Methods: Applications in Econometrics by H. D. Vinod 24. Identifying outliers and Influential Observations in Econometric Models by S. G. Donald and G. S. Maddala 25. Statistical Aspects of Calibration in Macroeconomics by A. W. Gregory and G. W. Smith


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26. Panel Data Models with Rational Expectations by K. Lahiri 27. Continuous Time Financial Models: Statistical Applications of Stochastic Processes by K. R. Sawyer

Volume 12. Environmental Statistics Edited by G. P. Patil and C. R. R a o 1994 xix + 927 pp.

1. Environmetrics: An Emerging Science by J. S. Hunter 2. A National Center for Statistical Ecology and Environmental Statistics: A Center Without Walls by G. P. Patil 3. Replicate Measurements for Data Quality and Environmental Modeling by W. Liggett 4. Design and Analysis of Composite Sampling Procedures: A Review by G. Lovison, S. D. Gore and G. P. Patil 5. Ranked Set Sampling by G. P. Patil, A. K. Sinha and C. Taillie 6. Environmental Adaptive Sampling by G. A. F. Seber and S. K. Thompson 7. Statistical Analysis of Censored Environmental Data by M. Akritas, T. Ruscitti and G. P. Patil 8. Biological Monitoring: Statistical Issues and Models by E. P. Smith 9. Environmental Sampling and Monitoring by S. V. Stehman and W. Scott Overton 10. Ecological Statistics by B. F. J. Manly 11. Forest Biometrics by H. E. Burkhart and T. G. Gregoire 12. Ecological Diversity and Forest Management by J. H. Gove, G. P. Patil, B. F. Swindel and C. Taillie 13. Ornithological Statistics by P. M. North 14. Statistical Methods in Developmental Toxicology by P. J. Catalano and L. M. Ryan 15. Environmental Biometry: Assessing Impacts of Environmental Stimuli Via Animal and Microbial Laboratory Studies by W. W. Piegorsch 16. Stochasticity in Deterministic Models by J. J. M. Bedaux and S. A. L. M. Kooijman 17. Compartmental Models of Ecological and Environmental Systems by J. H. Matis and T. E. Wehrly 18. Environmental Remote Sensing and Geographic Information Systems-Based Modeling by W. L. Myers 19. Regression Analysis of Spatially Correlated Data: The Kanawha County Health Study by C. A. Donnelly, J. H. Ware and N. M. Laird 20. Methods for Estimating Heterogeneous Spatial Covariance Functions with Environmental Applications by P. Guttorp and P. D. Sampson

852


21. Meta-analysis in Environmental Statistics by V. Hasselblad 22. Statistical Methods in Atmospheric Science by A. R. Solow 23. Statistics with Agricultural Pests and Environmental Impacts by L. J. Young and J. H. Young 24. A Crystal Cube for Coastal and Estuarine Degradation: Selection of Endpoints and Development of Indices for Use in Decision Making by M. T. Boswell, J. S. O'Connor and G. P. Patil 25. How Does Scientific Information in General and Statistical Information in Particular Input to the Environmental Regulatory Process? by C. R. Cothern 26. Environmental Regulatory Statistics by C. B. Davis 27. An Overview of Statistical Issues Related to Environmental Cleanup by R. Gilbert 28. Environmental Risk Estimation and Policy Decisions by H. Lacayo Jr.

Volume 13. Design and Analysis of Experiments Edited by S. G h o s h and C. R. R a o 1996 xviii + 1230 pp.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

The Design and Analysis of Clinical Trials by P. Armitage Clinical Trials in Drug Development: Some Statistical Issues by H. I. Patel Optimal Crossover Designs by J. Stufken Design and Analysis of Experiments: Nonparametric Methods with Applications to Clinical Trials by P. K. Sen Adaptive Designs for Parametric Models by S. Zacks Observational Studies and Nonrandomized Experiments by P. R. Rosenbaum Robust Design: Experiments for Improving Quality by D. M. Steinberg Analysis of Location and Dispersion Effects from Factorial Experiments with a Circular Response by C. M. Anderson Computer Experiments by J. R. Koehler and A. B. Owen A Critique of Some Aspects of Experimental Design by J. N. Srivastava Response Surface Designs by N. R. Draper and D. K. J. Lin Multiresponse Surface Methodology by A. I. Khuri Sequential Assembly of Fractions in Factorial Experiments by S. Ghosh Designs for Nonlinear and Generalized Linear Models by A. C. Atkinson and L. M. Haines Spatial Experimental Design by R. J. Martin Design of Spatial Experiments: Model Fitting and Prediction by V. V. Fedorov Design of Experiments with Selection and Ranking Goals by S. S. Gupta and S. Panchapakesan


853

18. Multiple Comparisons by A. C. Tamhane 19. Nonparametric Methods in Design and Analysis of Experiments by E. Brunner and M. L. Puri 20. Nonparametric Analysis of Experiments by A. M. Dean and D. A. Wolfe 21. Block and Other Designs in Agriculture by D. J. Street 22. Block Designs: Their Combinatorial and Statistical Properties by T. Calinski and S. Kageyama 23. Developments in Incomplete Block Designs for Parallel Line Bioassays by S. Gupta and R. Mukerjee 24. Row-Column Designs by K. R. Shah and B. K. Sinha 25. Nested Designs by J. P. Morgan 26. Optimal Design: Exact Theory by C. S. Cheng 27. Optimal and Efficient Treatment - Control Designs by D. Majumdar 28. Model Robust Designs by Y-J. Chang and W. I. Notz 29. Review of Optimal Bayes Designs by A. DasGupta 30. Approximate Designs for Polynomial Regression: Invariance, Admissibility, and Optimality by N. Gaffke and B. Heiligers

Volume 14. Statistical Methods in Finance Edited by G. S. M a d d a l a and C. R. R a o 1996 xvi + 733 pp.

1. Econometric Evaluation of Asset Pricing Models by W. E. Ferson and R. Jegannathan 2. Instrumental Variables Estimation of Conditional Beta Pricing Models by C. R. Harvey and C. M. Kirby 3. Semiparametric Methods for Asset Pricing Models by B. N. Lehmann 4. Modeling the Term Structure by A. R. Pagan, A. D. Hall, and V. Martin 5. Stochastic Volatility by E. Ghysels, A. C. Harvey and E. Renault 6. Stock Price Volatility by S. F. LeRoy 7. GARCH Models of Volatility by F. C. Palm 8. Forecast Evaluation and Combination by F. X. Diebold and J. A. Lopez 9. Predictable Components in Stock Returns by G. Kaul 10. Interset Rate Spreads as Predictors of Business Cycles by K. Lahiri and J. G. Wang 11. Nonlinear Time Series, Complexity Theory, and Finance by W. A. Brock and P. J. F. deLima 12. Count Data Models for Financial Data by A. C. Cameron and P. K. Trivedi 13. Financial Applications of Stable Distributions by J. H. McCulloch 14. Probability Distributions for Financial Models by J. B. McDonald 15. Bootstrap Based Tests in Financial Models by G. S. Maddala and H. Li

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16. Principal Component and Factor Analyses by C. R. Rao 17. Errors in Variables Problems in Finance by G. S. Maddala and M. Nimalendran 18. Financial Applications of Artificial Neural Networks by M. Qi 19. Applications of Limited Dependent Variable Models in Finance by G. S. Maddala 20. Testing Option Pricing Models by D. S. Bates 21. Peso Problems: Their Theoretical and Empirical Implications by M. D. D. Evans 22. Modeling Market Microstructure Time Series by J. Hasbrouck 23. Statistical Methods in Tests of Portfolio Efficiency: A Synthesis by J. Shanken

Volume 15. Robust Inference Edited by G. S. M a d d a l a and C. R. Rao 1997 xviii + 698 pp.

1. Robust Inference in Multivariate Linear Regression Using Difference of Two Convex Functions as the Discrepancy Measure by Z. D. Bai, C. R. Rao and Y. H. Wu 2. Minimum Distance Estimation: The Approach Using Density-Based Distances by A. Basu, I. R. Harris and S. Basu 3. Robust Inference: The Approach Based on Inftuence Functions by M. Markatou and E. Ronchetti 4. Practical Applications of Bounded-Influence Tests by S. Heritier and M-P. Victoria-Feser 5. Introduction to Positive-Breakdown Methods by P. J. Rousseeuw 6. Outlier Identification and Robust Methods by U. Gather and C. Becker 7. Rank-Based Analysis of Linear Models by T. P. Hettmansperger, J. W. McKean and S. J. Sheather 8. Rank Tests for Linear Models by R. Koenker 9. Some Extensions in the Robust Estimation of Parameters of Exponential and Double Exponential Distributions in the Presence of Multiple Outliers by A. Childs and N. Balakrishnan 10. Outliers, Unit Roots and Robust Estimation of Nonstationary Time Series by G. S. Maddala and Y. Yin 11. Autocorrelation-Robust Inference by P. M. Robinson and C. Velasco 12. A Practitioner's Guide to Robust Covariance Matrix Estimation by W. J. den Haan and A. Levin 13. Approaches to the Robust Estimation of Mixed Models by A. H. Welsh and A. M. Richardson


855

14. Nonparametric Maximum Likelihood Methods by S. R. Cosslett 15. A Guide to Censored Quantile Regressions by B. Fitzenberger 16. What Can Be Learned About Population Parameters When the Data Are Contaminated by J. L. Horowitz and C. F. Manski 17. Asymptotic Representations and Interrelations of Robust Estimators and Their Applications by J. Jureökovä and P. K. Sen 18. Small Sample Asymptotics: Applications in Robustness by C. A. Field and M. A. Tingley 19. On the Fundamentals of Data Robustness by G. Maguluri and K. Singh 20. Statistical Analysis With Incomplete Data: A Selective Review by M. G. Akritas and M. P. LaValley 21. On Contamination Level and Sensitivity of Robust Tests by J. Ä. Visgek 22. Finite Sample Robustness of Tests: An Overview by T. Kariya and P. Kim 23. Future Directions by G. S. Maddala and C. R. Rao

Volume 16. Order Statistics - Theory and Methods Edited by N. Balakrishnan and C. R. Rao 1997 xix + 688 pp.

1. Order Statistics: An Introduction by N. Balakrishnan and C. R. Rao 2. Order Statistics: A Historical Perspective by H. Leon Harter and N. Balakrishnan 3. Computer Simulation of Order Statistics by Pandu R. Tadikamalla and N. Balakrishnan 4. Lorenz Ordering of Order Statistics and Record Values by Barry C. Arnold and Jose A. Villasenor 5. Stochastic Ordering of Order Statistics by Philip J. Boland, Moshe Shaked and J. George Shanthikumar 6. Bounds for Expectations of L-Estimates by Tomasz Rychlik 7. Recurrence Relations and Identities for Moments of Order Statistics by N. Balakrishnan and K. S. Sultan 8. Recent Approaches to Characterizations Based on Order Statistics and Record Values by C. R. Rao and D. N. Shanbhag 9. Characterizations of Distributions via Identically Distributed Functions of Order Statistics by Ursula Gather, Udo Kamps and Nicole Schweitzer 10. Characterizations of Distributions by Recurrence Relations and Identities for Moments of Order Statistics by Udo Kamps 11. Univariate Extreme Value Theory and Applications by Janos Galambos 12. Order Statistics: Asymptotics in Applications by Pranab Kumar Sen 13. Zero-One Laws for Large Order Statistics by R. J. Tomkins and Hong Wang 14. Some Exact Properties Of Cook's DI by D. R. Jensen and D. E. Ramirez

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15. Generalized Recurrence Relations for Moments of Order Statistics from Non-Identical Pareto and Truncated Pareto Random Variables with Applications to Robustness by Aaron Childs and N. Balakrishnan 16. A Semiparametric Bootstrap for Simulating Extreme Order Statistics by Robert L. Strawderman and Daniel Zelterman 17. Approximations to Distributions of Sample Quantiles by Chunsheng Ma and John Robinson 18. Concomitants of Order Statistics by H. A. David and H. N. Nagaraja 19. A Record of Records by Valery B. Nevzorov and N. Balakrishnan 20. Weighted Sequential Empirical Type Processes with Applications to ChangePoint Problems by Barbara Szyszkowicz 21. Sequential Quantile and Bahadur-Kiefer Processes by Miklós Csörgö and Barbara Szyszkowicz

Volume 17. Order Statistics: Applications Edited by N. Balakrishnan and C. R. Rao 1998 xviii + 712 pp.

1. Order Statistics in Exponential Distribution by Asit P. Basu and Bahadur Singh 2. Higher Order Moments of Order Statistics from Exponential and Righttruncated Exponential Distributions and Applications to Life-testing Problems by N. Balakrishnan and Shanti S. Gupta 3. Log-gamma Order Statistics and Linear Estimation of Parameters by N. Balakrishnan and P. S. Chan 4. Recurrence Relations for Single and Product Moments of Order Statistics from a Generalized Logistic Distribution with Applications to Inference and Generalizations to Double Truncation by N. Balakrishnan and Rita Aggarwala 5. Order Statistics from the Type III Generalized Logistic Distribution and Applications by N. Balakrishnan and S. K. Lee 6. Estimation of Scale Parameter Based on a Fixed Set of Order Statistics by Sanat K. Sarkar and Wenjin Wang 7. Optimal Linear Inference Using Selected Order Statistics in Location-Scale Models by M. Masoom Ali and Dale Umbach 8. L-Estimation by J. R. M. Hosking 9. On Some L-estimation in Linear Regression Models by Soroush Alimoradi and A. K. Md. Ehsanes Saleh 10. The Role of Order Statistics in Estimating Threshold Parameters by A. Clifford Cohen 11. Parameter Estimation under Multiply Type-II Censoring by Fanhui Kong


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12. On Some Aspects of Ranked Set Sampling in Parametric Estimation by Nora Ni Chuiv and Bimal K. Sinha 13. Some Uses of Order Statistics in Bayesian Analysis by Seymour Geisser 14. Inverse Sampling Procedures to Test for Homogeneity in a Multinomial Distribution by S. Panchapakesan, Aaron Childs, B. H. Humphrey and N. Balakrishnan 15. Prediction of Order Statistics by Kenneth S. Kaminsky and Paul I. Nelson 16. The Probability Plot: Tests of Fit Based on the Correlation Coefficient by R. A. Lockhart and M. A. Stephens 17. Distribution Assessment by Samuel Shapiro 18. Application of Order Statistics to Sampling Plans for Inspection by Variables by Helmut Schneider and Frances Barbera 19. Linear Combinations of Ordered Symmetric Observations with Applications to Visual Acuity by Marlos Viana 20. Order-Statistic Filtering and Smoothing of Time-Series: Part I by Gonzalo R. Arce, Yeong-Taeg Kim and Kenneth E. Barner 21. Order-Statistic Filtering and Smoothing of Time-Series: Part II by Kenneth E. Barner and Gonzalo R. Arce 22. Order Statistics in Image Processing by Scott T. Acton and Alan C. Bovik 23. Order Statistics Application to CFAR Radar Target Detection by R. Viswanathan

Volume 18. Bioenvironmental and Public Health Statistics Edited by P. K. Sen and C. R. R a o 2000 xxiv + 1105 pp.

1. Bioenvironment and Public Health: Statistical Perspectives by Pranab K. Sen 2. Some Examples of Random Process Environmental Data Analysis by David R. Brillinger 3. Modeling Infectious Diseases - Aids by L. Billard 4. On Some Multiplicity Problems and Multiple Comparison Procedures in Biostatistics by Yosef Hochberg and Peter H. Westfall 5. Analysis of Longitudinal Data by Julio M. Singer and Dalton F. Andrade 6. Regression Models for Survival Data by Richard A. Johnson and John P. Klein 7. Generalised Linear Models for Independent and Dependent Responses by Bahjat F. Qaqish and John S. Preisser 8. Hierarchial and Empirical Bayes Methods for Environmental Risk Assessment by Gauri Datta, Malay Ghosh and Lance A. Waller 9. Non-parametrics in Bioenvironmental and Public Health Statistics by Pranab Kumar Sen

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10. Estimation and Comparison of Growth and Dose-Response Curves in the Presence of Purposeful Censoring by Paul W. Stewart 11. Spatial Statistical Methods for Environmental Epidemiology by Andrew B. Lawson and Noel Cressie 12. Evaluating Diagnostic Tests in Public Health by Margaret Pepe, Wendy Leisenring and Carolyn Rutter 13. Statistical Issues in Inhalation Toxicology by E. Weller, L. Ryan and D. Dockery 14. Quantitative Potency Estimation to Measure Risk with Bioenvironmental Hazards by A. John Bailer and Walter W. Piegorsch 15. The Analysis of Case-Control Data: Epidemiologic Studies of Familial Aggregation by Nah M. Laird, Garrett M. Fitzmaurice and Ann G. Schwartz 16. Cochran-Mantel-Haenszel Techniques: Applications Involving Epidemiologic Survey Data by Daniel B. Hall, Robert F. Woolson, William R. Clarke and Martha F. Jones 17. Measurement Error Models for Environmental and Occupational Health Applications by Robert H. Lyles and Lawrence L. Kupper 18. Statistical Perspectives in Clinical Epidemiology by Shrikant I. Bangdiwala and Sergio R. Mufioz 19. ANOVA and ANOCOVA for Two-Period Crossover Trial Data: New vs. Standard by Subir Ghosh and Lisa D. Fairchild 20. Statistical Methods for Crossover Designs in Bioenvironmental and Public Health Studies by Gail E. Tudor, Gary G. Koch and Diane Catellier 21. Statistical Models for Human Reproduction by C. M. Suchindran and Helen P. Koo 22. Statistical Methods for Reproductive Risk Assessment by Sati Mazumdar, Yikang Xu, Donald R. Mattison, Nancy B. Sussman and Vincent C. Arena 23. Selection Biases of Samples and their Resolutions by Ranajit Chakraborty and C. Radhakrishna Rao 24. Genomic Sequences and Quasi-Multivariate CATANOVA by Hildete Prisco Pinheiro, Frangoise Seillier-Moiseiwitsch, Pranab Kumar Sen and Joseph Eron Jr 25. Statistical Methods for Multivariate Failure Time Data and Competing Risks by Ralph A. DeMasi 26. Bounds on Joint Survival Probabilities with Positively Dependent Competing Risks by Sanat K. Sarkar and Kalyan Ghosh 27. Modeling Multivariate Failure Time Data by Limin X. Clegg, Jianwen Cai and Pranab K. Sen 28. The Cost-Effectiveness Ratio in the Analysis of Health Care Programs by Joseph C. Gardiner, Cathy J. Bradley and Marianne Huebner 29. Quality-of-Life: Statistical Validation and Analysis An Example from a Clinical Trial by Balakrishna Hosmane, Clement Maurath and Richard Manski 30. Carcinogenic Potency: Statistical Perspectives by Anup Dewanji


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31. Statistical Applications in Cardiovascular Disease by Elizabeth R. DeLong and David M. DeLong 32. Medical Informatics and Health Care Systems: Biostatistical and Epidemiologic Perspectives by J. Zvärovä 33. Methods of Establishing In Vitro-In Vivo Relationships for Modified Release Drug Products by David T. Mauger and Vernon M. Chinchilli 34. Statistics in Psychiatric Research by Sati Mazumdar, Patricia R. Houck and Charles F. Reynolds III 35. Bridging the Biostatistics-Epidemiology Gap by Lloyd J. Edwards 36. Biodiversity Measurement and Analysis by S. P. Mukherjee

Volume 19. Stochastic Processes: Theory and Methods Edited by D. N. Shanbhag and C. R. R a o 2001 xiv + 967 pp.

1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16.

Pareto Processes by Barry C. Arnold Branching Processes by K. B. Athreya and A. N. Vidyashankar Inference in Stochastic Processes by I. V. Basawa Topics in Poisson Approximation by A. D. Barbour Some Elements on Lévy Processes by Jean Bertoin Iterated Random Maps and Some Classes of Markov Processes by Rabi Bhattacharya and Edward C. Waymire Random Walk and Fluctuation Theory by N. H. Bingham A Semigroup Representation and Asymptotic Behavior of Certain Statistics of the Fisher Wrigh~Moran Coalescent by Adam Bobrowski, Marek Kimmel, Ovide Arino and Ranajit Chakraborty Continuous-Time ARMA Processes by P. J. Brockwell Record Sequences and their Applications by John Bunge and Charles M. Goldie Stochastic Networks with Product Form Equilibrium by Hans Daduna Stochastic Processes in Insurance and Finance by Paul Embrechts, Rüdiger Frey and Hansjörg Furrer Renewal Theory by D. R. Grey The Kolmogorov Isomorphism Theorem and Extensions to some Nonstationary Processes by Yüichirô Kakihara Stochastic Processes in Reliability by Masaaki Kijima, Haijun Li and Moshe Shaked On the supports of Stochastic Processes of Multiplicity Orte by A. Klopotowski and M. G. Nadkarni

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17. Gaussian Processes: Inequalities, Small Ball Probabilities and Applications by W. V. Li and Q.-M. Shao 18. Point Processes and Some Related Processes by Robin K. Milne 19. Characterization and Identifiability for Stochastic Processes by B. L. S. Prakasa Rao 20. Associated Sequences and Related Inference Problems by B. L. S. Prakasa Rao and Isha Dewan 21. Exchangeability, Functional Equations, and Characterizations by C. R. Rao and D. N. Shanbhag 22. Martingales and Some Applications by M. M. Rao 23. Markov Chains: Structure and Applications by R. L. Tweedie 24. Diffusion Processes by S. R. S. Varadhan 25. Itô's Stochastic Calculus and Its Applications by S. Watanabe

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