Multivariate T-Distributions and Their Applications

Multivariate t Distributions and Their Applications Almost all of the results available in the literature on multivaria...

Author: Samuel Kotz | Saralees Nadarajah

576 downloads 2030 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Multivariate t Distributions and Their Applications Almost all of the results available in the literature on multivariate t distributions published in the last 50 years are now collected together in this comprehensive volume. Because these distributions are becoming more prominent in many applications, this book is a must for any serious researcher or consultant working in multivariate analysis and statistical distributions. Much of this material has never appeared in book form. The first part of the book emphasizes theoretical results of a probabilistic nature. In the second part of the book, these are supplemented by a variety of statistical aspects. Various generalizations and applications

are dealt with in the final chapters. The material on estimation and regression models is of special value for practitioners in statistics and economics. A comprehensive bibliography of more than 350 references is included.

Samuel Kotz is Professor and Senior Research Scholar in the Department of Engineering Management and Systems Engineering at George Washington University. In addition to holding many distinguished vis-

iting positions at prominent universities, he has authored a number of books and more than 100 articles, is Editor-in-Chief and founder of the Encyclopedia of Statistical Sciences, and holds three honorary doctorates.

Saralees Nadarajah is Professor of Mathematics at the University of South Florida. He has made important contributions to distribution theory, extreme value theory and its applications in environmental modeling, branching processes, and sampling theory. He has authored more than 80 papers and two books.

Multivariate t Distributions and Their Applications Samuel Kotz George Washington University

Saralees Nadarajah University of South Florida

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge C132 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

O Samuel Kotz and Saralees Nadarajah 2004

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2004

Printed in the United States of America Typeface Computer Modern 10/13 pt.

System LAT C 26 [AU]

A catalog record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data Kotz, Samuel.

Multivariate t distributions and their applications / Samuel Kotz, Saralees Nadarajah. p. cm.

Includes bibliographical references and index. ISBN 0-521-82654-3

1. Multivariate analysis. 2. Distribution (Probability theory) I. Nadarajah, Saralees. 11. Title.

QA278.K635 2004 2003055353 519.5'35-dc2l ISBN 0 521 82654 3 hardback

Contents

List of Illustrations Preface

page ix xi

1 Introduction 1.1 1.2 1.3

1

Definition Representations Characterizations

1

2

1.4 A Closure Property 1.5 A Consistency Property 1.6 Density Expansions 1.7 Moments 1.8 Maximums 1.9 Distribution of a Linear Function

1.10 Marginal Distributions 1.11 Conditional Distributions 1.12 Quadratic Forms 1.13 F Matrix 1.14 Association 1.15 Entropy 1.16 Kullback-Leibler Number 1.17 Renyi Information 1.18 Identities 1.19 Some Special Cases

2 The Characteristic Function

7 8 9 9 10 12 15 15 16 19 20 20 21 23 26 30 33

2.1 2.2

Sutradhar's Approach Joarder and Ali's Approach

36 36 38

2.3

Levy Representation

41

3 Linear Combinations, Products, and Ratios 3.1 3.2 3.3

Linear Combinations Products Ratios

44

44 52 56

v

vi

Contents

4 Bivariate Generalizations and Related Distributions Owen's Noncentral Bivariate t Distribution Siddiqui's Noncentral Bivariate t Distribution Patil and Liao's Noncentral Bivariate t Distribution Krishnan's Noncentral Bivariate t Distribution Krishnan's Doubly Noncentral Bivariate t Distribution Bulgren et al.'s Bivariate t Distribution Siotani's Noncentral Bivariate t Distribution Tiku and Kambo's Bivariate t Distribution Conditionally Specified Bivariate t Distribution 4.10 Jones' Bivariate t Distribution 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

5 Multivariate Generalizations and Related Distributions Kshirsagar's Noncentral Multivariate t Distribution Miller's Noncentral Multivariate t Distribution Stepwise Multivariate t Distribution Siotani's Noncentral Multivariate t Distribution Arellano-Valle and Bolfarine's Generalized t Distribution Fang et al.'s Asymmetric Multivariate t Distribution Gupta's Skewed Multivariate t Distribution Sahu et al.'s Skewed Multivariate t Distribution Azzalini and Capitanio's Skewed Multivariate t Distribution 5.10 Jones' Skewed Multivariate t Distribution 5.11 Matrix-Variate t Distribution 5.12 Complex Multivariate t Distribution 5.13 Steyn's Nonnormal Distributions 5.14 Inverted Dirichlet Distribution 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

6 Probability Integrals Dunnett and Sobel's Probability Integrals Gupta and Sobel's Probability Integrals John's Probability Integrals 6.4 Amos and Bulgren's Probability Integrals 6.5 Steffens' Noncentral Probabilities 6.6 Dutt's Probability Integrals 6.7 Amos' Probability Integral 6.8 Fujikoshi's Probability Integrals 6.9 Probabilities of Cone 6.10 Probabilities of Convex Polyhedra 6.11 Probabilities of Linear Inequalities 6.12 Maximum Probability Content 6.13 Monte Carlo Evaluation 6.1 6.2 6.3

7 Probability Inequalities 7.1 7.2

Dunnett and Sobel's Probability Inequalities Dunn's Probability Inequalities

63 63 66 68 69 71 72 73 74 76 80

87 87 90 90 93 94 97 98 102 103 105 112 119 120 126 127 127 131 135 137 139 140 143 144 145 148 158 160 161 165 165 169

Contents

vii

7.3 Halperin's Probability Inequalities 7.4 Sidak's Probability Inequalities 7.5 Tong's Probability Inequalities

170

8 Percentage Points

Dunnett and Sobel's Percentage Points Krishnaiah and Armitage's Percentage Points Gupta et al.'s Percentage Points Rausch and Horn's Percentage Points Hahn and Hendrickson's Percentage Points Siotani's Percentage Points 8.7 Graybill and Bowden's Percentage Points 8.8 Pillai and Ramachandran's Percentage Points 8.9 Dunnett's Percentage Points 8.10 Gupta and Sobel's Percentage Points 8.11 Chen's Percentage Points 8.12 Bowden and Graybill's Percentage Points 8.13 Dunnett and Tamhane's Percentage Points 8.14 Kwong and Liu's Percentage Points 8.15 Other Results 8.1 8.2 8.3 8.4 8.5 8.6

9 Sampling Distributions

171

172

174 174 175 176

176 177 177 178

180 180 181

182

183 183

187 187 191

9.1

Wishart Matrix

191

9.2

Multivariate t Statistic

198

9.3 Hotelling's T2 Statistic 9.4 Entropy and Kullback-Leibler Number

10 Estimation 10.1 Tiku and Kambo's Estimation Procedure 10.2 ML Estimation via EM Algorithm 10.3 Missing Data Imputation 10.4 Laplacian T-Approximation 10.5 Sutradhar's Score Test 10.6 Multivariate t Model 10.7 Generalized Multivariate t Model 10.8 Simulation

11 Regression Models 11.1 Classical Linear Model 11.2 Bayesian Linear Models 11.3 Indexed Linear Models 11.4 General Linear Model 11.5 Nonlinear Models

12 Applications 12.1 Projection Pursuit 12.2 Portfolio Optimization

199

204 207

207 210 212

214 215

219 222 223 228 228

233 235 237 239 241

241

243

Contents

viii

12.3 Discriminant and Cluster Analysis 12.4 Multiple Decision Problems 12.5 Other Applications

244 245 246

References Index

247 269

List of Illustrations

1.1

1.2

1.3 1.4 1.5 1.6 3.1

Joint contours of (1.1) with degrees of freedom v = 1, zero means, and correlation coefficient p 0.8,0.6,.. . , -0.6, -0.8 page 3 Joint contours of (1.1) with degrees of freedom v = 2, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 4 Joint contours of (1.1) with degrees of freedom v = 10, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 5 Joint contours of (1.1) with degrees of freedom v = 30, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 6 Mutual information, (1.31), for p = 2 26 Mutual information, (1.31), for p = 4 27 Densities of the t-ratio distribution (3.11) for (mm, my, v) _ (0, 0, 1), (0, 0, 30), (1, 3, 1), (1, 3, 30), (3,1,1), (3, 1, 30), (3, 3, 1), and (3, 3, 30)

4.1

4.2

Jones' bivariate skew t pdf (4.33) for (a) vi = 2 and v2 = 3; and (b) vi = 2 and v2 = 20 Jones' bivariate skew t pdf (4.40) for (a) a = 1, b = 1, and

58

83

c = 1; (b) a = 3, b = 4, and c = 5; (c) a = 5, b = 1, and

c=1;and(d)a=1,b=5,andc=1 5.1

85

Fang et al.'s asymmetric t pdf (5.16) in the bivariate case

(a)m=2,mi =10,m2=10,andr12=0; (b)m=2,

m1 = 10, m2 = 2, and r12 = 0; (c) rn = 2, m1 = 10,

m2=10,and r12=0.5;and (d)m=2,m1=10,m2=10, 5.2

and r12 = 0.9 Jones' skewed multivariate t pelf (5.28) for p = 2 and (a)

a=6,v=3,andc=2; and (b) a = 2, v = 3, and c = 6 5.3 5.4

99 107

Jones' skewed multivariate t pdf (5.29) for p = 2 and (a) v = 1; and (b) v = 20 108 Jones' skewed multivariate t pdf (5.32) for p = 2 and (a) vo = 2, v1 = 4, and v2 = 4; (b) vo = 2, vi = 20, and v2 = 1; (c) vo = 2, vi = 1, and v2 = 20; and (d) vo = vi = v2 = 2 112

List of Illustrations 5.5

Steyn's bivariate pdf corresponding to (5.46) for t3 = 0 and (a) #1 = 0.8, n2 = -0.4, and r12 = 0.2; (b) i1 = 0.8, K2 = -0.4, and r12 = 0.8; (c) ic1 = -0.4, r-2 = 0.8, and r12 = 0.2; and (d) n1 = -0.4, K2 = 0.8, and r12 = 0.8

6.1

The sets Ar(c) and E(c) f1 {II z II= r} in two dimensions

124 146

Preface

Multivariate t distributions have attracted somewhat limited attention of researchers for the last 70 years in spite of their increasing importance in classical as well as in Bayesian statistical modeling. These distributions have been perhaps unjustly overshadowed - during all these years - by

the multivariate normal distribution. Both the multivariate t and the multivariate normal are members of the general family of elliptically symmetric distributions. However, we feel that it is desirable to focus on these distributions separately for several reasons: Multivariate t distributions are generalizations of the classical univariate Student t distribution, which is of central importance in statistical inference. The possible structures are numerous, and each one possesses special characteristics as far as potential and current applications are concerned.

Application of multivariate t distributions is a very promising approach in multivariate analysis. Classical multivariate analysis is soundly and rigidly tilted toward the multivariate normal distribution while multivariate t distributions offer a more viable alternative with respect to real-world data, particularly because its tails are more realistic. We have seen recently some unexpected applications in novel areas such as cluster analysis, discriminant analysis, multiple regression, robust projection indices, and missing data imputation. Multivariate t distributions for the past 20 to 30 years have played a crucial role in Bayesian analysis of multivariate data. They serve by now as the most popular prior distribution (because elicitation of prior information in various physical, engineering, and financial phenomena is closely associated with multivariate t distributions) and generate meaningful posterior distributions. This diversity and the apparent xi

xii

Preface

ease of applications require careful analysis of the properties of the distribution in order to avoid pitfalls and misrepresentation.

The compilation of this book was a somewhat daunting task (as our Contents indicates). Indeed, the scope of the multivariate t distributions is unsurpassed, and, although there are books dealing with multivariate continuous distributions and review articles in the Encyclopedia of Statistical Sciences and Biostatistics, the material presented in these sources is quite limited. Our goal was to collect and present in an organized and user-friendly manner all of the relevant information available in the literature worthy of publication. It is our hope that the readers - both novices and experts - will find the book useful. Our thanks are due to numerous authors who generously supplied us with their contributions and to Lauren Cowles, Elise Oranges and Lara Zoble at Cambridge University Press for their guidance. We also wish to thank Anusha Thiyagarajah for help with editing. Samuel Kotz Saralees Nadarajah

1

Introduction

1.1 Definition There exist quite a few forms of multivariate t distributions, which will be discussed in subsequent chapters. In this chapter, however, we shall describe the most common and natural form. It directly generalizes the univariate Student's t distribution in the same manner that the multivariate normal distribution generalizes the univariate normal distribution. A p-dimensional random vector X = (X1, ... , XP)T is said to have the p-variate t distribution with degrees of freedom v, mean vector µ, and correlation matrix R (and with E denoting the corresponding covariance matrix) if its joint probability density function (pdf) is given by

f(x) =

r ((v + p)/2) (xv)p/2r (v/2) IR,I1/2

1

T

1

Ll + v (x - µ) R- (x - µ)J

(v+P)/2

(1.1)

The degrees of freedom parameter v is also referred to as the shape parameter, because the peakedness of (1.1) may be diminished, preserved, or increased by varying v (see Section 1.4). The distribution is said to be central if µ = 0; otherwise, it is said to be noncentral.

Note that if p = 1, µ = 0, and R = 1, then (1.1) is the pdf of the univariate Student's t distribution with degrees of freedom v. These univariate marginals have increasingly heavy tails as v decreases toward unity. With or without moments, the marginals become successively less peaked about 0 e as v 11. If p = 2, then (1.1) is a slight modification of the bivariate surface of Pearson (1923). If v = 1, then (1.1) is the p-variate Cauchy distribution. If (v+p)/2 = m, an integer, then (1.1) is the p-variate Pearson type VII 1

2

Introduction

distribution. The limiting form of (1.1) as v -> oo is the joint pdf of the p-variate normal distribution with mean vector p and covariance matrix E. Hence, (1.1) can be viewed as an approximation of the multivariate

normal distribution. The particular case of (1.1) for µ = 0 and R = Ir is a mixture of the normal density with zero means and covariance matrix vIr - in the scale parameter v. The class of elliptically contoured distributions (see, for example, Fang et al., 1990) contain (1.1) as a particular case. Also (1.1) has the attractive property of being Schurconcave when elements of R satisfy ri.7 = p, i $ j (see Marshall and Olkin, 1974). Namely, if a and b are two p-variate vectors with compo> bp, and nents ordered to achieve al > a2 > > ap and b1 > b2 > , p - 1 and ai < E if this ordering implies bi for k = 1,2,.. . 1 1 EP 1 ai < EP bi, then (1.1) satisfies f (a) > f (b). In Bayesian analyses, (1.1) arises as: (1) the posterior distribution of the mean of a multivariate normal distribution (Geisser and Cornfield, 1963; see also Stone, 1964); (2) the marginal posterior distribution of the regression coefficient vector of the traditional multivariate regression model (Tiao and Zellner, 1964); (3) the marginal prior distribution of the mean of a multinormal process (Ando and Kaufman, 1965); (4) the marginal posterior distribution of the mean and the predictive distribution of a future observation of the multivariate normal structural model (Fraser and Haq, 1969); (5) an approximation to posterior distributions arising in location-scale regression models (Sweeting, 1984, 1987); and (6) the prior distribution for set estimation of a multivariate normal mean (DasGupta et al., 1995). Additional applications of (1.1) can be seen in the numerous books dealing with the Bayesian aspects of multivariate analysis.

1.2 Representations If X has the p-variate t distribution with degrees of freedom v, mean vector µ, and correlation matrix R, then it can be represented as If Y is a p-variate normal random vector with mean 0 and covariance

matrix E, and if vS2/Q2 is the chi-squared random variable with degrees of freedom v, independent of Y, then

X = S-1y + µ.

(1.2)

This implies that X I S = s has the p-variate normal distribution with mean vector µ and covariance matrix (1/82)E.

1.2 Representations

3

WAM

rha:OAM:1

M

-3

4

4

4

-1

-1

0

1

0

1

2

2

3

3

4

4

4

4

-1

0

0

2

1

1

2

3

3

4

4

4

4

-1

-1

0

0

1

1

2

3

2

3

2

3

t*-04L1 M

N

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

4

4

-1

0

1

Fig. 1.1. Joint contours of (1.1) with degrees of freedom v = 1, zero means, and correlation coefficient p = 0.8,0.6..... -0.6, -0.8

Introduction

4

M

N

0

-3

-2

-1

0

1

2

3

-3

4

4

-1

0

1

0

I

-3

-2

W.W

dMW

4

-1

2

3

4

4

-1

0

1

-1

0

1

2

3

2

3

2

3

rt*-01n 2

2

3

4

4

-1

0

1

b.4u2

ft:-0.4p4

n

n

-3

3

-1

0

1

2

3

4

4

-1

0

1

2

3

4

-2

-1

0

1

Fig. 1.2. Joint contours of (1.1) with degrees of freedom v = 2, zero means and correlation coefficient p = 0.8, 0.6, ... , -0.6, -0.8

5

1.2 Representations

rha'6 Vlo

3

rla

-1

ri

nth10

4

rl

3

-2

-1

rl

-0.4,rtw10

3

3

-1

0

1

W,4,rIU.-1O

2

3

3

3

0,0, eo

0

1

rl

2

3

3

2

1

-1

0

2

3

3

-2

-f

1

2

3

-0,41t10

0

rtftA

-0,6pit10

0

-1

0

1

2

3

to

1

2

3

Fig. 1.3. Joint contours of (1.1) with degrees of freedom v = 10, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8

Introduction

6

w4a

4

4

-1

0

1

W.W

2

3

-3

4

-1

0

1

W.W

2

3

4

-2

-1

0

1

2

3

2

3

2

3

r io:41A 30

4

4

-2

4

-1

-1

0

0

1

1

2

2

3

3

4

4

4

4

-1

-1

0

0

1

1

2

2

3

3

4

4

4

4

-1

0

rh

-l2 w

-1

0

1

1

Fig. 1.4. Joint contours of (1.1) with degrees of freedom v = 30, zero means, and correlation coefficient p = 0.8,0.6,..., -0.6, -0.8

1.3 Characterizations

7

If V'/2 is the symmetric square root of V, that is,

V1/2V1/2 = V ,,, Wp (R-1, v + p - 1) ,

(1.3)

where Wp(E,n) denotes the p-variate Wishart distribution with degrees of freedom n and covariance matrix E, and if Y has the p-variate normal distribution with zero means and covariance matrix vIp (Ip is the p-dimensional identity matrix), independent of V, then

X=

1Y

(V1/2)

+µ

(1.4)

(Ando and Kaufman, 1965). This implies that X I V has the p-variate normal distribution with mean vector µ and covariance matrix vV-1.

1.3 Characterizations From representation (1.2) it easily follows for any a ; 0 that X has the joint pdf (1.1) if and only if X I S2 = s2 ' N (µ, s-2 E) .

q

(aTEa)

-112

aT (X - µ) I S2 = s2 - 1 / 2 aT

N (0,8-2)

(X - µ) ,., t,,,

and this is one of the earliest characterization results given in Cornish (1962). This result can also be obtained by using the representation (1.4): X has the joint pdf (1.1) if and only if

XIV - N(p,vV-1)

(aTEa)-1/2 aT (X - µ) I V - N (0, v

C*

(aTEa) -1/2 aT (X - µ) ,., tv,

(aTV- 'a) / (aTEa))

as noted by Lin (1972). Lin (1972) obtained two further characterizations using the representation (1.2). Let vS2 _ x2 and let X1, X2, ... , Xp be conditionally independent continuous random variables symmetrically distributed with E(Xk S2 = s2) = µk and Var(Xk S2 = s2) _ 0.2./s2 < oo for k = 1,. .. , p. Then the following characterizations are valid I

I

(X1, X2, ... , Xp)T has the joint pdf (1.1) with mean vector µ, covari-

ance matrix D, and degrees of freedom v if and only if

(Xk - µk)2 k=1

PL72

Fp,v

Introduction

8

where D is a p x p diagonal matrix with its kth diagonal element equal to 01'

In the special case Q2 = 01 2 for all k and the conditional pdf of Xk S2 = s2 is positive and differentiable for all x E R, (X1, X2i ... , X ,)T has the joint pdf (1.1) with zero means, covariance matrix a2IP, and degrees of freedom v if and only if the joint pdf of X1, X2,. .. , Xr is a function of x1 + x2 + + xP only.

1.4 A Closure Property Consider Studentizing transformations T : In -4 Ik, depending on matrices A(n x k), B(n x v) and 1Z(n x n), given by T (X)

_ fATX

1IB XII

such that ATfIB = 0. Jensen (1994) established that the class of multivariate t distributions is closed under the transform T(.). Specifically, assume ATA = Ik, BTB = I, and X is distributed according to (1.1) with zero means, correlation matrix In, and degrees of freedom m. Under these assumptions, Jensen showed that T(X) is also distributed according to (1.1) with zero means, correlation matrix Ik, and degrees of freedom v.

Jensen (1994) also studied the concentration properties of (1.1) via peakedness by varying its parameters. If X is multivariate normal, then the transformation X -4 T(X) diminishes the peakedness. If, on the other hand, X is distributed according to (1.1) with mean vector µ1n, covariance matrix o 2In, and degrees of freedom m, then the transformation is peakedness-enhancing for all m < v. If m > v > 2, then

the transformation serves to increase variances. For any m > v > 0 the marginal distributions are less peaked after T(X) than before in the sense of Birnbaum (1948). If m = v, then the marginals are identical before and after T(X), thus exhibiting identical tail behavior. If v > m then marginals are more peaked (in the sense of Birnbaum, 1948) after applying T (X) than before; and if v > m > 2, then T (X) serves as a variance-diminishing transformation.

1.5 A Consistency Property

9

1.5 A Consistency Property A random vector X = (X1, ... , XP) T is said to have the spherical distribution if its joint pdf can be written in the form

where g(.) is referred to as the density generator. The p-variate t pdf (1.1) with it = 0 and E = Ip is spherical because in this case, 1 + u -(p+v)/2

F ((v +p)/2)

=

9(u)

(V)

(xv)P/2F (v/2)

Other examples of spherical distributions include the multivariate normal and the multivariate power exponential. A spherical distribution is said to possess the consistency property if

r

P+1

O

P

xi p dxp+1

g

xz P

g

(1.6)

i=1

(i=1

for any integer p and almost all x E RP. This consistency property ensures that any marginal distribution of X also belongs to the same spherical family. Kano (1994) provided several necessary and sufficient conditions for a spherical distribution to satisfy (1.6). One of the them is that g must be a mixture of normal distributions; specifically, there exists a random variable Z > 0, unrelated to p, such that, for any p,

= f(-_)

.f (u I P)

z

p

l

exp (__2 ) F(dz),

where F(.) denotes the cumulative distribution function (cdf) of Z. Since the multivariate t is a mixture of normal distributions (see (1.2)), it follows that it must have the consistency property. Other distributions that have the consistency property include the multivariate normal and the multivariate Cauchy. Distributions that do not share this property include the multivariate logistic, multivariate Pearson type II, multivariate Pearson type VII , and the multivariate Bessel.

1.6 Density Expansions Fisher (1925) and later Dickey (1967a) provided expansions of the pdf f (x)

-

1 + x2

F((v + 1)/2) 7rvF (v/2)

1.

V

(v+1)/2

}

Introduction

10

of the univariate Student's t distribution. The expansion in the latter paper involves Appell's polynomials, and hence recurrence schemes are available for its coefficients. Specifically, 00

f(x) =

exp ( 12

Qk )k=0 >

(_

vxz)

12

(1 + v),

/f

where k-1

Qk(t) = Pk(t) -

EQ,(t)Pk-l(r) 1=0

Here, Pk(t) are polynomials (in powers of t) satisfying

Pk(t)(1 +

v)-k

=

11- 1+

v)

exp(-t)

and Pk(F) denotes the polynomial Pk(t) with the powers t' replaced by F(r + 1/2). Dickey (1967a) also provided an analog of (1.7) for the multivariate t pdf (1.1). It takes the same form as (1.7) with x 2 replaced by (x- µ)T R-1(x - µ), v+ 1 replaced by v + p, and with (1.8) replaced by 1

(t)

=

Pk (t)

kQk -1

- F(P/2) EQc (t)Pk-c (rr),

where FP indicates the substitution of F(r + p/2) for t'.

1.7 Moments Since Y and S in (1.2) are independent, the conditional distribution of (Xi, Xj), given S = s, is bivariate normal with means (µi, µ0, common variance a2/s2, and correlation coefficient rid. Thus, E (Xi)

= E [E (Xij S = s)] E (pi) µi

To find the second moments, consider the classical identity Coo (Xi, Xj)

= E [Cov (Xi, Xj) I S = s] +Cov [E (Xil S = s) E (Xjj S = s)]

1.7 Moments

11

for all i, j = 1, ... , p. Clearly, one has E [Cov (Xi, Xj) I S = s]

= o2rzjE

SZ 1)

and

Cov [E (XZIS = s) E (Xj S = s)] =

0.

If v > 2, then E(1/S2) exists and is equal to v/{o2(v - 2)}. Thus, by choosing i = j and i < j, respectively, one obtains

Var(X )

=

V

v - 2

and

Cov (Xi, Xj) =

v

v

2rZj.

Hence the matrix R is indeed the correlation matrix as stated in definition (1.1). In the case where µ = 0, the product moments of X are easily found by exploiting the independence of Y and S in (1.2). One obtains J"'rl,rz,..., rp j=1

E S-r (llYr3)] j-1 1

Q-rvrl2E II Yr'JI E j-1 provided that r

[xV r]

+ rp < v12. In the special case where

r1 + r2 +

Y1,. .. , Yp are mutually independent, one obtains p

,Url,rz,..., rp

Q-rvr/2E [XV r] II E [Yjr'] j=1

If anyone of the rj's is odd, then the moment is zero. If all of them are even, then r12fjp=1 llri,rz,...,rp

=

(v - 2)(v - 4) ... (v - r)

1)}

Introduction

12

In particular, v

=

µ2,0,...,o

v > 2,

v - 2' 3v2

4,

(v - 2)(v - 4) µ2,2,0,...,o

v2

-

(v

v > 4,

- 2)(v - 4)'

and

=

/12,2,2,0,...,0

v3

v>6.

(v - 2)(v - 4)(v - 6)'

1.8 Maximums Of special interest are the moments of Z = max(Xi,... Xp) when XT = (X1,. .. , Xp) has the t pdf (1.1) with the mean vector µ and ,

covariance matrix E. These moments have applications in decision theory, particularly in the selection and estimation of the maximum of a set of parameters. It also has applications in forecasting. The problem of finding the moments of Z has been considered by Raiffa and Schlaifer (1961), Afonja (1972), and Cain (1996). Raiffa and Schlaifer (1961) provided an expression for E(Z - 0) for

the case where p = 3 and µ = 01p (where 1p denotes a vector of l's). Afonja (1972) generalized this for the general case of unequal means, variances, and correlations. We mention later a particular case of this result for µ = 01p. Let Op(y; R) denote a p-dimensional normal pdf with zero means, unit variances, and correlation matrix R. Also let Ri denote a p x p matrix with its (j, j')th element equal to ri,jj,, where ri,jj, (j, j' 54 i) is the correlation between (Xi, -Xj) and (Xi, -Xj,) and ri,ij = corr(Xi, Xi - Xj). Then the kth moment of Z is given by E (Zk)

_

1 p k kl Bk-7 ( r (v12)( \2(v - 2) (j/ j=0 2_1

where

µj (yi)

=J 0

foo"

r

... J0

I

))'/2

V2

) µ (yi)

yi 0p (y; Ri)

0 dypdyp-1...dyi...dy2dy1

(1.10)

1.8 Maximums

13

is the marginal moment (up to a constant) of truncated normal variates.

The mean and variance can be derived easily from this formula. For example,

E(Z) = 9+{E(W) -9}F (-±)

/r (2)

,

where W = max(Yi, ... , Y,) for a p-variatenormalll random vector yT = (Y1, ... , YY) with means equal to 9 and covariance matrix (vl (v - 2)) E. Afonja (1972) showed further that n

E(W)

+

Vvv

2

iipi (yi) i1 -

where pi (yi) is given by (1.10) for j = 1. More recently, Cain (1996) considered two forecasts Fl and F2 of a future variable Y where the forecast errors Xl = Fi -Y and X2 = F2 - Y are assumed to have the bivariate t distribution with means (A1, A2), variances (o , o ), correlation coefficient p, and degrees of freedom v >

2. Cain was interested in the maximum Z = max(Xi, X2) of the two forecast errors and whether this nonlinear function could be useful as a component of a linear combination forecast. It was shown that the pdf of Z can be written as the sum =

f(z)

fi(z) + f2 (Z),

where

f, (z)

=

r

vv2tu \ vv 2aipZ/ 1+vlz-`k

l+

L

xT

ok

z=& l

- p a;

J

1-p2 v-2+ (z° )

2

for k = 3 - j, j = 1, 2. Here, t and T are, respectively, the pdf and the cdf of the Student's t distribution with degrees of freedom v. Integration by parts yields that

E(Z) = pi

J

fl(z)dz + p2

f2(z)dz +

FOO 00

Var(Z)

= Ql F fl(z)dz 00

f2(z)dz

a2 J 00

T

,

Introduction

14

ao

+ (µl - µ2 )2

fl(z)dzj

J

f2(z)dz

ao

µl

(µl - p2) tv_2

T

00

µ2

J

/

-T (µl - µ2)

C µ1

T

µ'2

f2(z)dz 00

fl z dz

J

(µl-µ2)(Q2-Q1µl-µ2+

T(v - 2) t

T

and 00

Cov (Z, XI) = Q1 f7fl(z)dz+P0'0`2f f2(z)dz 00

0

+ (µl

- µ2)(o -

µ1 - µ2

(

T(v - 2)

T

)

where T = o uu2 - 2pala2. The two integrals in the above expressions can be evaluated as 00

Al

7'v

fl(z)dz

A2

v-2

C

0J0

and

f2(z)dz

=

1 -T (µ1 - /t2

v

T

0o

2

The expression for Cov(Z, X2) can be obtained by switching the subscripts 1 and 2. As v -> oo, the above expressions can be reduced by replacing by and respectively. On the other and extreme, as v -+ 2+, the expressions could be reduced by using the fact

that

IxI

ifx=0,

0,

lim ti_2(x) v*2+v-2

1/2, ifx54 0,

and

1/2,

ifx > 0, ifx = 0,

0,

ifx a, and denotes the digamma function. For further discussion of maximum entropy methods, see Fry (2002).

1.16 Kullback-Leibler Number The mutual information of a continuous random vector X with joint pdf f (x) and marginal pdfs f (xi), i = 1, ... , p is defined by T (X)

= E [_log{

f(X) ... f

f (xl)

l

(xp) }]

(1.30)

24

Introduction

with the domain of variation given by 0 < T(X) < no. (The reader should not confuse this with the transformation T(X) given in (1.5).) The quantity (1.30) can be considered a measure of dependence (Joe, 1989). The larger the T(X), the higher the dependence among the variables Xi, i = 1,.. . , p. Naturally, T (X) = 0 implies that the variables are independent; this latter statement follows from the fact that T is a special case of the Kullback-Leibler number, KL(f, g) (Kullback, 1968). When the variables of X are multivariate normal with covariance matrix E, it is easy to compute T(X) as the difference between entropies given

by (1.28); specifically,

T(X;E) = H(X;E)-H(X;D), where D is a diagonal matrix corresponding to E with the elements au,... , app This is due to the well known fact that uncorrelatedness implies independence in the normal case. This fact also implies that T(X; I) = 0. In general, for any member of an elliptical family of distributions, this is not true; in other words, uncorrelatedness does not imply that T(X) = 0. The mutual information attempts to summarize in a single number the whole dependence structure of the multivariate distribution of X. Guerrero-Cusumano (1996b) derived the form of (1.30) for the multi-

variate t distribution. For a central p-variate t, it turns out that

=

T (X)

Il - 2 log JRI

(1.31)

,

where 1 is given by

Q=

(i

tog

r (p/2) BP xp/2 B (p+v a) 2

' 2P_)

+ Al + v) 2

p2v{0(p2v)-

\2/}.

{ (1

2

v)

-

(2 (1.32)

It is easy to see that fl -4 0 as v -4 no. The mutual information for the multivariate normal distribution with correlation matrix R is given by -(1/2) log I R I (Kullback, 1968). The particular case of (1.31) for v = 1 gives the mutual information for the multivariate Cauchy distribution with c taking the simpler form Il

=

log

8P r(p+2) }-12pt,,(12p)- (2)}. /2

(12)

ll JJJ

1.16 Kullback-Leibler Number

25

Table 1 in Guerrero-Cusumano (1996b) provides values of (1.32) for a range of v and p. The following is an abridged version.

Constant 52 for T(X) = 52 - (1/2) log I R I V

p=1

p=2

p=3

p=4

p=5

1

0

0.4196180

0.949615

1.530690

2.141170

2

0

0.2927000

0.705474

1.184010

1.704100

3

0

4

0

0.2254360 0.1835450

0.975130 0.832265

1.431820 1.240460

5

0 0

7

0

8

0

9

0

10

0

0.727338 0.646600 0.582368 0.529959 0.486337 0.449434

1.096790

6

0.565424 0.473177 0.407380 0.357917 0.319304 0.288289 0.262813 0.241503

0.1548760 0.1339950 0.1180970 0.1055830 0.0954730 0.0871342

0.984235 0.893344 0.818244 0.755056 0.701101

Figures 1.5 and 1.6 graph T(X) in (1.31) for p = 2 and p = 4, respectively. The correlation matrix R is taken to have the equicorrelation structure ri1 = p, i # j. It is interesting to see the "dale-shaped" threedimensional plot. The figures show that, as one moves toward the center of the "dale," the dependence among the variables decreases, and, as one moves away from the center, the dependence increases. For the normal case, Linfoot (1957) and Joe (1989) suggested a parameterization for T(X) to make it comparable to a correlation coefficient.

They defined the induced correlation coefficient based on the mutual information as PI

=

1 -exp {-2T (X)}.

(1.33)

Guerrero-Cusumano (1998) suggested a similar measure for the multivariate t distribution referred to as the dependence coefficient. It is given by

pI

=

1- I R I exp (-252).

(1.34)

The dependence coefficient is a quantification of dependence among the p variables of X. This follows from the fact that independence implies

Introduction

26

Fig. 1.5. Mutual information, (1.31), for p = 2

pi = 0 and that T(X) = oo implies pI = 1. When v - oo, (1.34) coincides with (1.33). The sampling properties of (1.31) will be discussed in Chapter 9.

1.17 Renyi Information Since the concept of Renyi information is not widely available in the literature, we provide here a brief discussion of the concept. Renyi information of order A for a continuous random variable with pdf f is defined as ZR(A)

1 1

log

(ffk(x)dx)

(1.35)

1.17 Renyi Information

27

rho

Fig. 1.6. Mutual information, (1.31), for p = 4 for A , 1. Its value for A = 1 is taken as the limit

IR(l)

lim IR(A)

A-+1/

- J f(x)log(f(x))dx

-E [log f(X)], which is the well known Shannon entropy. Renyi's (1959, 1960, 1961) generalization of the Shannon entropy allows for "different averaging of probabilities" via A. Sometimes (1.35) is also referred to as the spectrum of Renyi information. Renyi information finds its applications as a measure of complexity in areas of physics, information theory, and engineering to describe many nonlinear dynamical or chaotic systems (Kurths et al., 1995) and in statistics as certain appropriately scaled

Introduction

28

test statistics (Renyi distances or relative Renyi information) for testing hypotheses in parametric models (Morales et al., 1997). The gradient I'R(A) = dIR(A)/dA also conveys useful information. In fact, a direct

calculation based on (1.35) - assuming that the integral f f'(x)dx is well defined and differentiation operations are legitimate - shows that

(1 - a) f f 'f ZR(1) _ lim A-+1

f),\lo(g

+ log

(ffA(x)dx)]

I

f f A(x) loge f (x)dx

1 lim

f f (x)dx

2 A->i

-C

f f (x)1og f (x)dx)

2

)

f f (x)dx

-1 V ar [log f (X )J . In other words, the gradient of Renyi information at A = 1 is simply the negative half of the variance of the log-likelihood compared to the entropy as the negative of the expected log-likelihood. Thus, the variance of the log-likelihood I f := 21I(1) measures the intrinsic shape of the distribution. This can be seen by observing that If, where f (x) = (1/Q)g((x - p)/Q). In fact, according to Bickel and Lehmann (1975), it can serve as a measure of the shape of a distribution. In the case where f (x) has a finite fourth moment, it plays a similar role as a kurtosis measure in comparing the shapes of various frequently used densities and measuring the heaviness of tails, but it measures more than what kurtosis measures. Renyi information of order A for a p-variate random vector with joint pdf x is defined as IR(A)

:=

log Y f' (x1,. .. , xp) dxl ... dxp 1

1 1

.

(1.36)

The gradient IR(A) and the measure If are defined similarly. Song (2001) provided a comprehensive account of IR(A), IR(A), and If for well known univariate and multivariate distributions. For the univariate Student's t distribution with degrees of freedom v, it can be shown for A > 1/(1 + v) that IR(A)

1

-A

to

B ((vA

g

t

1)/2,1/2)

B A(v12,112)

+ 2 log(y),

1.17 Renyi Information IR(A)

-

29

{ B ((vA + A - 1)/2,1/2)

log

B (v/2,1/2)

+(1-A)2(1+v)

(VA + A-1)

,

- (1-A)(1+v) ((1+v)A)]/(1A)2 and

(1+v)2 {(v)(i+u)}

If (V)

Using tables in Abramowitz and Stegun (1965), one obtains the particular values 2 *7r

if (1)

3'

If (2)

9-

If ( 3

37x2

4

=

)

=

If (4)

47x2

3

- 12

775 36

=

1f (5)

'

37x2 -

,

257r2 12

115 4

It is interesting to note that the measure If (v) decreases as v increases, which makes sense since the tails become lighter as v increases. In fact, it can be shown, using asymptotic formulas for the trigamma function,

that

If(v) = 1/2, which corresponds to the measure If(v) for

the normal distribution. For the central p-variate t distribution with correlation matrix R and degrees of freedom v, it can be shown for A > p/ (p + v) that ZR(A)

=

1 1

log { B ((vA+

p/2),p/2) l

p A2

J

+ 2 log {(v7r) ' I R

- logy (P2) 1

log

B ((vA +pA - 1)/2,p/2) B (v/2, p/2)

+(1 - A)(p+v), (vA+A_\ 2

Introduction

30

((p +2v)\ 1

(1 - A) (p + v)

I/

2

(1 - a)2,

and

Zf(v)

_

(p4v)2

l l Ml

(±±)}.

For p = 1, these expressions reduce to those derived for the Student's t distribution.

1.18 Identities In one of the earliest papers on the subject, Dickey (1965, 1968) provided two multidimensional-integral identities involving the multivariate

t distribution. This first identity expresses a moment of a product of multivariate t densities of the form (1.1) as an integral of dimension 1 less than the number of factors. Consider the product K

g (x)

= [I

[1

+ (x - µk)T Rk (x - µk)

Uk/2

(1.37)

k=1

where each Rk > 0 and Uk > 0, and so each term may not have a finite integral. The identity seeks an expression for the complete p-dimensional integral of s g, where s(x) is a polynomial in the coordinates of x. Let Y be a p-variate normal random vector with the covariance matrix and mean vector given by -1

K

Du 1

=

(>ukRk) k=1

and

µu = D'

K

ukRkµk1 k=1

respectively.

Ek 1 CkUk

For given constants Ck > 0, k = 1, ... , K, let u. = and uk = Vku.. Then the quantity defined by N8lu =

E(s(Y)) can be expanded as a polynomial in 1/u. as N8Iu

= E hj(v1i ... , vK) u7

1.18 Identities

31

Given this terminology, the identity can now be expressed as

s(x)g(x)dx fn P

2-jr

KO

X

(V.2 p

ID,I-1/2h.7 (v1i...,VK)

-j/ J

K

Wv-(- -p)12dv1 ... dvK-1,

K

(1.38)

k=1

where K

Ko =

v/2 7r

fl r (vk/2), k=1

V.

K _ E vk, k=1

K

Dv =

k=1

vkRk,

1

K

WV

= E Vk

K

T

K

f, + /lk Rkltk llI -

vkRkµk

Dv 1

vkRkILk

(k=1

k=1

k=1

and o, is the simplex K

1

Q = {(v1. .. , 4JK) : 1: Ckvk = 1,

vk> ()

k=1

This identity has applications to inference concerning the location parameters of a multivariate normal distribution. In the particular case K = 2, Rk = ryklp, and s =- 1, (1.38) reduces to

f

g (x) dx =

Cy2U.-p)12

B

Rp

Vi

xF1 (

v v2

\2

2

v. - p v. - p v.

2

2

'

2

where C

xpl2

IF ((v. - p) /2)

r(v1/2) r (v2/2)

vi/2 v2/2, 71

'Y2

2,

z1, z2

,

(1.39)

Introduction

32

F1 is Appell's hypergeometric function of two variables defined by F1 (a;

_ _

Q'; 7; x, y) 1

1

B(a,7-a)

to-1(1

-

t)7-a-1(1

-

tx)-a(1

- ty)-R'dt

o

(1.40)

(see, for example, Erdelyi et al., 1953), and z1 and z2 are the two real roots of the equation

z2+(7211 µ2-µl112+ryi-llz-7211 µ2-µ11I2 =

0.

The integral (1.39) is proportional to a multivariate generalization of the Behren-Fisher density. For an asymptotic expansion of (1.37) in powers of vk, see Dickey (1967a). The second identity given by Dickey (1968) - see also Dickey (1966b)

- expresses the density of a linear combination of independently distributed multivariate t vectors as an integral of dimension 1 less than the number of summands. Consider the r-variate vector 6 formed by the linear combination K

= E BkXk,

6

k=1

where Xk are independent qk-variate standard t random vectors with zero means, covariance matrix I9k , and degrees of freedom vk. Dickey (1968) showed that b has the representation S

=

vkU; 1BkBkY, k=1

where Uk are independent chi-squared random variables with degrees of freedom vk and Y is an independent r-variate standard normal vector. As a consequence, 6 has the further representation K k=1

vk, Vk = Uk/ >K1 U, and W is an independent where v. = 1 r-variate standard t vector with degrees of freedom v.. If the matrix E BkBk is nonsingular, the distribution of 6 is nondegenerate with the

1.19 Some Special Cases

33

joint pdf K

=C

f (S)

I

-(v.+r)/2

K

ft

1 + ST

(vk/vk) BkBk

k=1

k=1

K

(vk/vk) BkBT dv1 ... dvK-1,

(1.41)

k=1

where

r ((v. + r) /2)

C = 7.r/2r (v1/ 2) ... r (vK/2) and as above K or

=

{(vi. ..,VK) : Y Ckvk = 1,

Vk > 0

k=1

This identity has applications in Behrens-Fisher problems. The version

of(1.41)forK=2andBk=/3kis f (S)

= CB

(v1 + p v2 + p1 \\

2

xF1

J

2

'

v1+p v.+p v.+p V. 2

2

,

,

2

, 2 +A Z1, z2

where

C-

r((v. +p)/2) 7r n/2r (v1/2) ... r (v2/2)

(v202)-(U1+n)/2,

(v1Q2)vl/2

F1 is Appell's hypergeometric function as defined in (1.40), and z1 and z2 are the two real roots of the /equatioon z2 + r II S I2 + v2#22

V, 012 V2 #22

1Iz.1

II S II2

=

0.

1'2182

This special case is essentially equivalent to the two-factor version of (1.38). Moreover, (1.41) is a generalization of Ruben's (1960) integral representation (in the univariate case) for the usual Behrens-Fisher densities.

1.19 Some Special Cases A number of special cases of (1.1) have been studied in the literature with great detail. Cornish (1954), in his early paper, considered the

Introduction

34

special case of (1.1) when µ = 0 and R is given by the equicorrelation matrix 1

-1/p

R =

-1/p ... -1/p ... -1/p 1

-1/p -1/p ...

1

/

The following interesting properties were established

XTR-1X has the noncentral F distribution with degrees of freedom p and v. XTR-1 X has the Fisher's z distribution with degrees of freedom p - q and v - when X is subject to the linearly independent homogeneous conditions represented by the equation SX = 0, where S is of order

gxpandrank q Q, i=1

where Ai are the roots of the equation I AR-1 - A I= 0 or, alternatively, the latent roots of the matrix RA. Consequently, the distribution of XT AX is Fisher's z with degrees of freedom q and v if and only if the nonzero latent roots of RA are all equal to unity. If the distribution of X is partitioned as in (1.11)-(1.13), then

E (X1 I X2) = -R111R22x2, and

Var (X1 I X2) =

V + x2 (R22 - R21R111R12) x2

v+p-pl-2

In the particular case pi = 1,

r

E(X1 I X2) _

2T xj, j=2

-1

1.19 Some Special Cases Var (X1 I X2)

_

(p+1)v

35 3

p

2p(v + p - 3) + 4(v + p - 3) j=2 =2

p+1

+2(v+p-3) E[Var(X1 I X2)] = Var (X2)

=

EX'xk'

v p+1

v2 2p v

v 2822 V

i = 2, ... , p. - p(v - 2)' Furthermore, the residual variance of Xl with respect to X2 is Cov (X1i Xi)

v p+l

v-2 2p

'

and the partial correlation coefficient of X1 with respect to X2 is

-1/2. Patil and Kovner (1968) provided a detailed study of the trivariate t density

f (x1,x2,x3)

F((n + 3)/2) (nr')3/2 1 - p2r (n/2) X

Cl +

-(n+3)I2 1x1 - 2px1 + x22+ x223 I n 1-p2

Among other results, Taylor series expansions - in powers of 1/n - of the density and associated probabilities in rectangles were given.

2

The Characteristic Function

The characteristic function (cf) of the univariate Student's t distribution for odd degrees of freedom was derived by Fisher and Healy (1956). Ifram (1970) gave a general result for all degrees of freedom, but Pestana

(1977) pointed out that this result is not quite correct. More recent derivations are presented in Drier and Kotz (2002). Here we discuss two independent results on the characteristic function for the multivariate t distribution. The first one, due to Sutradhar (1986, 1988a), provides

a series representation for the cf while the other, due to Joarder and Ali (1996), derives an expression in terms of the MacDonald function. The expressions given are rather complicated; thus, further research and possible simplifications may be desirable.

2.1 Sutradhar's Approach Let X be distributed according to

f (x)

=

F'1 v +p)12)

(irv)P/ r (v12) IRI 1/2

1 + 1 (x - µ)T R (x - µ)1

-(,+n)/2

J

(2.1)

Consider the transformation Y = R-1/2(X - µ). It then follows that the joint pdf of Y is

=

vv/212) r(v12)

36

(v I

\

P

+

E Jk

k=1

(v+n)l2

2.1 Sutradhar's Approach P

F ((v +p)/2)

exp (itT y)

v-v/2r (v12)

37 (v+P)/2

v+Ey

dy1 dye ... dyP

k=1

To evaluate this integral, Sutradhar (1986) makes the orthogonal transformation Y = I'Z, where the first column of the p x p matrix r is the vector t2

t1

tp

IltIIIItIII

'Iltll

tTt. It follows that the cf of Z is given by

with 11 t 11=

Oz (t; v)

/2 F

=

vV

x

v ((r (

p) 2) f exp (i II t II z1) dzi

f f ... f (cp + z2)-(v+p)/2 dz2 ... dzp, (2.2)

where zk E R, k = 1,. .. , p and cp = v+ EP-1 zk. Successive integration of (2.2) with respect to zp, z _ 1 , . .. , z2 yields

=

4z (t; v)

L,/2(( v + 1)/2) J1 vPr (v/2)

(2.3)

where 00

J1

=J

-(v+1)12

exp (z II t ll w) (+u2 + v)

dz.

(2.4)

00

Note that J1 is an improper integral along the real axis, where w denotes a complex variable. For odd v, the integrand has poles of integer order

while, for fractional and even v, the poles are of fractional order. Sutradhar (1986) evaluated J1 separately for the three cases: odd v; even v; and fractional v, using the relations that

= exp (itT µ) OY R' /2t- v

Ox (t; v)

(2.5)

and

OY (t; v)

= Oz (t; v)

(2.6)

to obtain the following expressions for Ox. For the case of odd v V17r

x(t; v) =

F ((v + 1)/2) exp (itT, 2"-1r (v/2)

X k=1

-

vtTRt)

(2. - k - 1 (2 vtT Rt) (k-1)! Tn, - k

k-1


38

where m = (v + 1)/2. For the case of even v (-1)m+1I'

_

OX (t; v)

((v + 1)/2) exp (itT ft)

V,7rI' (v/2) r', (n - k + 1/2) x

00

vtT Rt

1

iz n-0 (n) +

2n

2

f1(n - j) (log vtTRt 4 j-0

m-1

m-1

j=O

k=o>koj

11

(n - k)

- 1"(n + 1) I'(n+1))

(2.7) '

where m = v/2. Finally, when v is of fractional order OX (t;

v)

-

7r(-1)mv(v/z)-mr ((v + 1)/2) exp (itTµ)

1/2) r' ( - k) zn vtTRt 21uko1(n--k)

2T (v/2)

1

1

X

n-0

n

- (tTRt)

Uer(n + 1 -

2

!

IT- of (n - k)

(2.8)

where m = [(v + 1)/2] is the integer part of (v + 1)/2 and _ (v/2) - m is such that 0 0 and nonintegral positive values of a is

r2j-a

°°

°°

Ka(r) = 2`1 j=0 j!(1 - a)j4j

+ 2-(1+0')

1`

r2j+a

j=0 j!(1 + a)j4i

(2.11)

(see, for example, Spainer and Oldham, 1987, Chapter 51), where (c)j = c(c+1) . . . (c+j-1) denotes the ascending factorial. Using (2.9), Joarder and Ali (1996) rewrote the integral (2.4) as J1

f00

=

{cos (II t II w) + i sin (II t II w)} ao

2v-(v+1)/2 f

/

CIO

cos (II t II W)

w22

v

2

dz

1+1 w2)-

2

C1 +

v

\

o

\C1 +

dz

F(11 t

2 v/z

Kv/z (tip II t

(v + I

II)

.

Thus, using (2.3), one obtains 11

Oz (t)

=

2u

/2 r(v12)

(II ,/it

Hence, using the relationships (2.5) and (2.6), one arrives at the expression for the cf given by Joarder and Ali (1996)

Ox(t)

= exp (itT µ) 2I v/2vRt('I 2) Kv/z (II

t II)

(2.12)

Joarder and Ali also provided expansions of this cf using the series representations (2.10) and (2.11). For positive and even v, applying (2.10), one obtains v/2-1

ox (t)

= exp (itT µ)

Ci (j) II

vRt I I2j

j=0 co

+

1: C2 (j) 11 IVRt III+2j j=0


40

00

- E C3 (j) II

vRt IIv+2j log (II v t II)

j=0

where

Ci(j) = C2(j)

-

(-1)1 (v/2 - j - 1)! (v/2 - 1)!j!4j

0(1+j)+0(1+v/2+j)+log4 2v (v/2 - 1)!j! (v/2 + j!) 4i

and 21-v4-i

(v/2 - 1)! (v/2 + j)! For positive and odd or fractional v, applying (2.11), the cf (2.12) becomes 00

Ox (t)

exp (itTµ) j=0

{Di(j) IIt II2j +D2(j) II

vRt IIv+2j}

,

where 4-j

Di(j) =

j!(1 - v/2)j

and 2-v4-3F (-v/2)

D2(j)

F (v/2) (1 + v/2) j

Since the univariate Student's t, multivariate Cauchy, and Pearson's type VII are all particular cases of (2.1), the corresponding cfs in terms of the MacDonald function can be obtained from (2.12). They are as follows

For the univariate Student's t distribution with the pdf jr

v7rF (v/2)

(1+v)/2

1 + x2

F ((1 + y)/2)

(

v

)

(where x E IR and v > 0), the cf in terms of the MacDonald function is

t Ox (t)

=

2v/2

41II, (v/2) Kv/2

(compare with Dreier and Kotz, 2002).

(v'V I t

2.3 Levy Representation

41

For the p-variate Cauchy distribution with the joint pdf

r ((1 +p)/2)

(x)(1+p)/2 IR,I1/2 [1 + (x

f (x)

- µ)T R-1 (x - µ)] - (1+p)/2

(where x E WP), the cf is Ox (t)

= exp (itTµ)

2/;r II vt II1/2 K1/2 (II vt II)

= exp (it T p- II Aft II) it/(2r) exp(-r)

which follows by using the result that K1/2(r) = (see, for example, Tranter, 1968, page 19).

For the p-variate Pearson type VII distribution with the joint pdf

f (x)

=

r(m)

xp/2r (m - p12) IRI 1/2

11

+

(x - µ)T

R-1 (x

- µ)]

_m

v (where x E RP, m > p/2 and v > 0), the cf becomes

II t IIm-p/2 = exp (itTµ) 2m,-p/2-1r (m - p/2) Km-p/2 v

Ox (t)

II

vRt II

.

2.3 Levy Representation Infinite divisibility of the univariate Student's t distribution was first proved by Grosswald (1976) - see also Kelker (1971) for a partial result. Later, Halgreen (1979) established the Levy representation of its cf. For a multivariate t, Takano (1994) provided the first proof of infinite divisibility and the corresponding Levy representation. Consider the standard

case µ = 0 and R = Ip. In this case, after suitable transformation, the joint pdf (2.1) can be written in the form

r (m r (/))

f (x) =

(1+ II x

112)-(m+p/2)

The corresponding cf is

21-r` 0 (t)

=

r(m) IItIImKm (IItII)

.

(2 . 13)

Takano (1994) derived the Levy representation of (2.13) in the form (t)

= exp f.,

lexp (itTx) -1II X lip

itTX 1+ 11 X II2


42

x U000 gm(2w)Lp/2 ( 2w II X 11) dw} dxl ... dxr where x

La(x) =

(2ir)-axaKa(x),

f2aI (a + 1/2)

J. (x)

'2

1

z2)a-1/2

(1 -

exp (ixz) dz,

and

=

Ya(x)

1

sin (a7r)

{cos (air) Ja(x) - J-a(x)}

.

Note that JJ(.) and

are the Bessel functions of the first and second kinds, respectively, of order in. Now consider (2.13) itself as a joint pdf fm (x)

(2.14)

= C 11 x 11m K. (11 X Il)

where the normalizing constant C is 21-m-p/2 r, = (2ir)p/21' (m + p/2)' Using properties of the MacDonald function Ka(), (2.14) can be reduced to the simpler forms f1/2 (X)

= n

fn+1/2 (X)

2eXp(-IIx2kkn(n )k)

=

11Xlln-k'

k=0

and

n-I f. (X)

=

C2'"-1

E

((Akn

- 1)!

11

X

112k

k=0 00

1

r11X IIl

k!(n+k)! (\

2(n-1-k)

2

x{log("2 -)-1,P(1+k)-' (1+n+k)}.

2.3 Levy Representation

43

Takano (1994) established further that the joint pdf (2.14) is also infinitely divisible and that its cf (t)

_ (1+ II t

II)-(m+p/2)

admits the Levy representation fi (t)

= exp

[(2m + p)

f {exp (itTx) P

where Lp/20 is as defined above.

-1}

Lp/2 (11 11)

11 x lip

dxl

.

dxp]

3

Linear Combinations, Products, and Ratios

3.1 Linear Combinations The distribution of linear combinations of independent t random variables has been studied by numerous authors, among them Fisher (1935), Chapman (1950), Fisher and Healy (1956), Ruben (1960), Patil (1965), Ghosh (1975), and Walker and Saw (1978). Johnson and Weerahandi (1988) tackled the distribution of linear combinations for multivariate t random vectors. Their results are included here for completeness and to motivate further multivariate extensions. We hope that our readers will benefit from studying this material, which contains fruitful ideas and also refers to the original papers for further details. Chapman (1950) considered the difference D = X1 - X2 of two independent t random variables Xj with common degree of freedom v. If v is odd, then it is known that the characteristic function of Yj = Xj// is

0,(t) = E [exp (itYj)]

/exp (- I t I)

(v-1)/2

((v - 1)/2 + k)! 12t 1-2 ' -k v!((v - 1)/2 - k)! V

2s-1F(v/2)

k=O

(3.1)

Using this representation, Chapman provided the following general ex-

pression for the pdf of W = D/'/ f 00

f (w)

=

2J

exp {-i(v + 1)t} 02(t)dt, V 00

which may be integrated to obtain the pdf of D in a closed form for values such as v = 1, 3, 5, and so on. Chapman tabulated the distribution of D

for v = 1, 3,5,7,9, 11. 44

3.1 Linear Combinations

45

Fisher and Healy (1956) considered the mixture D = a1X1 + a2X2 of two independent t random variables Xj with degrees of freedom vj when aj > 0, j = 1, 2. It is obvious that the characteristic function of D is the product

c,, (al It 1)0v2(a2It1), is as defined in (3.1). Since the product is simply a polynomial in t of degree (vl + 112)/2 - 1, it can be expanded in a finite series of terms of in which the highest value of m = vl + 112 - 1. For example, in the special case 111 = 3 and 112 = 5, one can write where

exp {(sin 9 + cos 9) t} 03 (sin Bt) 05 (cos Bt)

25/tang (V/-5

+

+

7sin9+

-,,f3- tan 0) 3

5v/5- -10ytan0+9ftan2B

(V+ 'tan0)3

7cos9 t [(V/Jsin8+ 05

cos9)t,

3

+

(_'Jtan9

03

[(V3- sin9+ti5- cos9)tJ;

from this one can easily deduce the pdf of D. Ruben (1960) provided results on the distribution of D = X1 sin 0 X2 cos 0, when Xj are independent t random variables with degrees of freedom vj and 0 is a fixed angle between 0 and 90 degrees. This statistic was originally proposed by Fisher (1935) as the basis for testing or estimating the difference in means of two unconnected and totally unknown normal populations, the "fiducial distribution" of the difference between the latter quantity and the corresponding sample mean difference, when suitably standardized, being supposed to be that of the statistic D. Ruben obtained the pdf of D in the integral form

rl

f(d) =

f

{

1

vl + v2B((vl + 112)/2,1/2)

d202(S) }_(V1+V2+1)/2

1+

111 + 112

- s)(v2/2)-1 ds, x(s) S(vl/2)-1(1 B (vl/2,112/2)

where

0(s)

=

(U1 + 112)8(1 - s)

u1(1 -s)sin20+v2scos29

(3.2)


46

It follows directly from (3.2) that D may be expressed in the form D

X

=

0(S)'

where X is a Student's t random variable with degrees of freedom v1 +

v2 and S is a Beta variable with parameters v1/2 and v2/2, that is, a variable with pdf given by the second term under the integral sign in (3.2), with the first term under the integral sign representing the conditional pdf of X/O(S) for fixed S. In the special case vl = V2 = v and 6 = 45 degrees, (3.2) reduces to f(d)

- ev

F(v + 1/2)F((v + 1)/2)

1

F2(v/2)F(v/2 + 1)

2

2

d2

v + 1. v +

(v +

Fl

2

2

I;-

2v

where 2F1 denotes the Gauss hypergeometric function. By using the appropriate four of the group of 24 transformations of the hypergeometric function 2F1 (see, for example, Whittaker and Watson, 1952, page 284), the above pdf may be expressed in the following three additional ways

f(d)

2 F(v + 1/2)F((v + 1)/2) V

F2(v/2)F(v/2 + 1) x 2F1 (

f (d)

_

2

(i+) 2v1-v d2

1 v+2 d2 '2' 2 ' 2v)

2 F(y + 1/2)F((y + 1)/2)

V-v r2(v/2)r(v/2 + 1)

(v+1/2)

d2

(1 + 2v)

1 v+2 (v+,;;d22), d2

1

X 2F1

(3.3)

'

and

f (d)

=

(u+1)/2 /-2F(v + 1/2)F((v + 1)/2) d2 F2(v/2)F(v/2 + 1) (1 + 2v) VV

x 2F1

(

l+v 1-v v+2 2

'

2

'

2

d2

' d2 +

2v) (3.5)

Note that (3.3) and (3.5) may each be expanded as a terminating series (refer to the definition of the Gauss hypergeometric function) when v is odd, and also that (3.4) is expressed as the product of the pdf of a t random variable with degrees of freedom 2v and the Gauss hypergeometric function.


47

In the special case v1 = no and v2 = v, (3.2) reduces to

f(d)

;v+2' 2 sin

F(v)F((v + 1)/2) (v + 1 IF, 2 27r sin OF(v/2)F(v + 1/2)

=

1

d2 z

o

where 1 F1 is the confluent hypergeometric function. Using Kummer's

first transformation for the confluent hypergeometric function, one can obtain the alternative form d2 d F(v)F((v + 1)/2)

fO

-

27r sin BF(v/2)F(v + 1/2) 1

V

x 1F1

(2;v+

exp

C- 2 sinz 6

d2

2; 2sin2 9)

Ruben (1960) also provided expressions for the cdf of D, but these were infinite series involving incomplete gamma function ratios. For tables of percentage points of D, see Sukhatme (1938), Fisher and Yates (1943), and Weir (1966). Ghosh (1975) provided explicit expressions for the cdf in terms of simple hypergeometric functions when D = X1-X2 and X3 are independent Student's t random variables with common degree of freedom v < 4. In particular, Ghosh obtained the following expressions for Pr(0 < D < d) 7r

1

arctan (d) 2

{(1 + q)E(q) - (1 - q)K(q)} ,

2/d (18 + dz) 7r (12 + d2)2

1

+

arctan

(d1

and

1 (8p4 -3 1P3 + 48p2 + 5p + 2) E(p)

64p

- (4p4 - 16p3 + 6p2 + 4p + 2) K(p)

,

for v = 1, v = 2, v = 3, and v = 4, respectively, where p = d2/(16+d2),

q=d2/(8+d2), n/ z

E(x) =

1 - x sinz sds fo

48


is the complete elliptical integral of the second kind, and 7r/2

K(x) =

ds

1-xsin2s

Jo

is the complete elliptical integral of the first kind. Similar expressions

for Pr(0 < D < d) as a finite sum of terms can be obtained for any positive integer v. Ghosh also provided a tabulation of the numerical values of Pr(D < d) for v = 1(1)20 and d = 0.0(0.5)10.0. Walker and Saw (1978) expressed the cdf of the linear combinations of any number of Student's t random variables with odd degrees of freedom as a mixture of t distributions. Define the linear combination as n

D=E

ak

vkXk,

k=1

+ an = 1 and Xk are independent Student's t

where ak > 0, a1 +

random variables with degrees of freedom vk = 2mk+1, Mk = 0, 1, 2, .... Construct a matrix Q whose element in row i and column j is the coefficient of exp(- I t I) t Ii in 02i+1(t) (see equation (3.1)), that is, I

2ji! 2i-j ! (2z)I'Iz j)

I

0

'f -0 1 2 if j > i.

-0 1 2

The characteristic function of D when vk = 2mk + 1 can be written as n

fi(t) = E [exp (itD)] = 1 10,, (akt)

,

k=1

and since exp(I t I)i(t) is a polynomial in I t I, one may obtain a vector A such that fi(t)

= exp(- ItI)

AkItIk. k-o

Walker and Saw (1978) showed that the cdf of D can expressed as the weighted sum

s Pr(D < d) =

E%Hk(d),

k=0

where 77

T = xTQ-1

,


49

n

S=

Mk, k=1

and

= Pr [X2k+1
v2 > 2. Based on a t-approximation proposed by Patil (1965), an improved approximation is

Tv O

Pr (D < d)

where T is the cdf of the Student's t distribution of v degrees of freedom,

v = 4+

cost y2 + sing V, vi-2 V2-2

2

SIn4 V1

COSQ i/2

(vz-2)

(v

(where vl > v2 > 4), and h2

v

v-2

cost v2

(v2-2

Sin2 U

+

v1-2)'

where v1 > v2 > 4. Ghosh (1975) derived another approximation that requires only tables of the normal distribution Pr(D < d)

-

(D

(d) - dO(d/y) IQ, (d) + Q2 (d) + 32i

V2

v2

Q3 (d) v2


50

where

= (1+A)(d2+10),

Q1(d)

=

Q2 (d)

1 + A2 384

(3d6 + 98d4 + 620d2 + 168)

+64 (d6 - 1044 + 3642 -456),

Qs (d)

=

1+

A2

24576

(d10 + 66d8 + 1016d6

+A(1 + A) (3d10 24576

- 1296d4 - 65328d2 -141408)

- 58d8 - 280d6 + 6864d4 - 70032d2)

1277A(1 + A)

+ 256 and A = v2/vl. Ghosh showed evidence to suggest that this is far more accurate than Patil's approximation. Johnson and Weerahandi (1988) considered linear combinations of t random vectors in a Bayesian context. Suppose Yii,... , Y1,m1 and Y21, . , Y2M2 are independent samples from two p-variate normal populations N(µ1, E1) and N(µ2, E2), respectively, where the population covariance matrices are unknown and not necessarily equal. Let Yi and Si denote the corresponding sample mean vectors and sample covariance matrices. Johnson and Weerahandi considered the distribution of the quadratic form

Q=

(6 - d)T

V-1

(S - d)

,

where S = Al - µ2, d = Yl - Y2, and V is any p x p positive definite matrix. Note that µi - Yi have the central p-variate t distribution with covariance matrix Si/(mi - p) and degrees of freedom mi - p. Under the diffuse prior distribution

p(µ1,El,µ2) E2) _

IE1EiI,

the posterior cdf of Q can be expressed as 00

F(q)

nq

_ E E (wj) Fp+2j,n 10(p+2j)] ,

(3.7)

j=o

where n = m1 +m2 - 2p and the expectation is taken with respect to the


51

beta random variable B, which is distributed as Be((ml - p)/2, (m2 p)/2). The wj are defined in terms of 0 (an arbitrary constant) and )j by the recursive relation 1 r-1

Wr = 2r E Hr-jwj, j=0

where wo

Hr

j Ir

=

e

j=1

and Aj are the ordered eigenvalues Al < ... < A of the matrix

m1BV-1/2S1V-1/2 + 1122(11V-1/2S2V-1/2. B) In the particular case V = cS1 + (1 - c)S2, the Aj can be conveniently obtained by using the relation

m2(1 - B) + m1m2B(1 - B) {c+ (1 where 1, ... , p are the eigenvalues of ST 1S2. In the univariate case,

(3.7) reduces to give the posterior cdf of Y = (Al - µ2) - (-ti - x2) as

-m2 -2 -1i+ =1 1-B (_ymi ml B M2 mz where s1 and s2 are the sample variances and the expectation is taken with respect to B, which is distributed as Be((ml - 1)/2, (m1 - 1)/2).

The result (3.7) can also be used to deduce the pdf of U = (T1 ± T2)T(T1 ± T2), where the Ti are independent p-variate random vectors having the t distribution with covariance matrix (ai/mi)Ip and degrees of freedom mi. It turns out that the cdf of U is

F(u)

r

= E ILFP,ml+mz

u(ml+m2) p

B(1-B) a1 + B (a2 - ai)

l

where the expectation is taken with respect to B, which is distributed


52

as Be(mi/2,m2/2). Johnson and Weerahandi also established several interesting bounds on (3.7), one upper bound being F(q)

< E Fp,_

p m1BS1 + m2(11- B)S2

V-1/2SiV-1/2. Furthermore, it was shown that similar rewhere Si = sults hold if the diffuse prior in (3.6) is replaced by the natural conjugate prior distribution.

3.2 Products The distribution of products of Student's t random variables has been studied by Harter (1951) and Wallgren (1980). Products of Student's t random variables arise naturally in classification problems. In many educational and industrial problems it may be necessary to classify persons or objects into one of two categories - those fit and those unfit for a particular purpose. In formulating a classification problem, assume that for p tests one knows the scores of N1 individuals known to belong to population 11, and of N2 individuals known to belong to population II2, along with those of the individual under consideration, a member of the population H, where it is known a priori that 11 is identical with either I'll or 112. Assume further that the distribution of the test scores of the individuals making up 11, and 112 are two p-dimensional normal distributions, which possess the same covariance matrix but are independent of each other. In order to classify the individual in question into either 11, or 112i Wald (1944) introduced the statistic V given by P

P

V =

ij S Yi,n+1Yj,n+2,

i=1 j=1

where n = N1 + N2 - 2, sij is the (i, j)th element of the inverse of the matrix S defined by 1

sij

n

= n E Yi,kYj,k, k=1

and Yi,k (i = 1 , ... , p; k = 1, ... , n + 2) are iid normally distributed random variables with unit variances and expected values E(Yi,k) = 0, k = 1,... , n, E(Yi,n+1) = µi, and E(Yi,n+2) = µ2

3.2 Products

53

In the particular case p = 1, (3.8) can be written as

V=

Yi,n+l_i,n+2 Z

where

Z =

Yz

2

k=1

Thus, one sees that the V in (3.9) is a product of two independent Student's t variates. Harter (1951) derived the exact distribution of this V. In particular, he showed that the pdf of I V I is given by 00

1

f (I V I)

_ err (n/2) xr2

n/2+j

j=O

1:(-1) n

Iv

(1+j+n/2)

(2+2i+n)/r(li)

ifIvl>n/2, and by f (I V I)

=

exp (-µ2/2) 00

xr2

IV

r(1 + j)

7rf' (n/2)

I-(1 {+n/2)

(2+2j+n) - (2µ2)kr (k+ 2+n+2jl 4

/f k (2k)!

4

J

ifIvl 0, w < 0, where

irQv (9; P, w)

v sin 0 sin (0 + A)

=

( w + v sin 0 sin (0 + A) )

a = arctan -

v/2 7

1 - P2

P

and angle A is defined by sin A = 1 - p2, cos A = p for 0 < A < ir. The corresponding pdf f (w) can be obtained by differentiation. The pdf has a singularity at w = 0 and, considered as a function of p and w, f (w; p) = f (-w; -p). The limit of f (w) as p -* 1- is the F1,,, density. Moreover, if pl < P2, then F(w; pi) > F(w; p2) for any w. In the noncentral case, the cdf F(w) is given by

F

=

f f o

J

h(s) exp a2

(-v2/2) 27r

1 x' (WS 2/(v+A2)-Al-PV dvds 1-p2 J

3.2 Products

55

or, equivalently, 00

F

f -A2 h(s) exp (-y2/2)

J

o

27f

o0

+ A2) + A1 +pvl

J

X (-ws2/(v 1-p2

duds

where \i = µi/a, i = 1, 2 are the noncentrality parameters and vv/2sv-1 exp (-vs2/2) 2(v-2)/2F (v/2)

The two double integrals above are bounded above by 4)(A2) and 4)(-A2), respectively. As a function of Al and A2, F has the following properties:

F(w;A1,A2) = F(w;A2,A1), F (w; -Ai, -A2) = F (w; A1, A2)

F(w;a1,-A2) = F(w;-A1,A2), lim F (w; cA2, A2)

= 0 if c > 0,

A2--*00

lim F (w; cA2i A2)

A2- 00

lim F (w; cA2i A2) A2-*00

=

1/2 if c = 0,

= 1 if c < 0,

lim F (w; A1, B) _ 4)(-B),

A1-*00

and

lim F (w; a1i B) =

ai-r-00

(P(B).

Also, for Al > 0, A2 > 0, w > 0, and -1 < p < 0, F(w; A1, A2) is a strictly decreasing function of Al and A2. Thus, the maximum of F(w) over the region Al > 0, A2 > 0 occurs at Al = A2 = 0. Since (X, Y) and S2 are independent, the first two moments of W are given by

E(W) = E (XY) E (1/S2) and

E (W2)

= E (X2Y2) E (1/S4)

.


56

It is known (see, for example, Craig, 1936) that E (XY) = Q2 (A1.A2 + P)

,

2A2 E (X2Y2) = Q4 (A2 +1 A2 + 2 4pAIA2 + A 1 2

+1+2 P2 )

,

and

-

E

v,r (y/2 - i)

L(1/S2)iJ

(2a2)Z P (v/2)

forv>2'. 3.3 Ratios

For a bivariate t random vector (X1, X2) with degrees of freedom v, the mean vector (mi, my) and correlation matrix I2 define the ratio W = X1/X2. The distribution of this ratio is of interest in problems in econometrics and ranking and selection. Press (1969) derived the pdf of W as

f(w) = 1+w2

+

k2q

1

{2T Cq

V

q*

+1

I -1}

-oo<w-Ar L1

a 0

1

11

i6 0 0

J

-2

0

2

4

T---T---l a

-2

0

Fig. 3.1. Densities of the t-ratio distribution (3.11) for (my, my, v) = (0,0, 1), (0, 0, 30), (1, 3, 1), (1, 3, 30), (3,1,1), (3, 1, 30), (3, 3, 1), and (3, 3, 30)

3.3 Ratios

59

Percentage points w for (m, my) = (0, 0) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

2

-31.820 -31.820

-6.314 -6.314

-3.078 -3.078

3.078 3.078

6.314 6.314

31.820 31.820

5

-31.820

-6.314

-3.078

3.078

6.314

31.820

-31.820 -31.820

-6.314 -6.314

-3.078 -3.078

3.078 3.078

6.314 6.314

31.820 31.820

10

30

Percentage points w for (my, my) = (1, 3) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

-10.254 -5.791 -1.331 -1.041 -0.681

-1.794 -0.902 -0.382 -0.312 -0.256

-0.721 -0.357 -0.166 -0.135 -0.109

1.321

1.120

2.394 1.795

10.853 6.842

1.018

1.486

0.951

1.288

0.922

1.211

6.464 2.892 2.350

2

5 10

30

Percentage points w for (m,,, my) = (3,1) v 1

2 5

10

30

p=0.01 p=0.05 p=0.1 p=0.9 p=0.95 p=0.99 -51.268 -58.076 -64.675 -67.570 -69.742

-8.970 -10.187 -11.427 -11.984 -12.406

-3.604 -4.034 -4.524 -4.756 -4.934

6.604 7.337

8.007 8.293 8.505

11.970 13.422 14.774 15.355 15.788

54.267 61.290 67.984 70.894 73.074

Percentage points w for (m, my) = (3, 3) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

-12.970 -8.172 -2.668 -0.118 0.130

-1.852 -0.474 0.229 0.334 0.384

-0.442 0.183 0.425 0.477 0.505

2.242 2.141 2.038 1.986 1.943

3.652 3.205 2.802 2.627 2.497

14.770 11.028 7.498 6.019 4.997

2 5

10

30


60

Since it is evident from (3.11) that f (w) -* 0 rapidly as m,, and my become large, values of my, my greater than 3 are not considered. The t-ratio distribution has one or two modes, depending upon the values of the parameters. The location of these modes are solutions of the equation v + 1 I + A(w)

T,+1 C q*

/

14

= B(w),

v+3I

(3.12)

\ q*

where

_

it (W)

1

B(w) = 2 +

(v+1)q2 qw+mx l+w2 (q*)2

(v + 2)F (v/2)

A(w)

-+W2

3qw + rn

1

{

q

}

2F ((v + 1)/2) m1 + w2 + 3qw

2

+ r ((v + 2)/2)

fl' ((v + 1)/2) x

1 + w2

{ 1 + (q/q*)2 }

(v+2)/2

q* {m 1 + w2 + 3qw}

{w 1+w2 -q'mx

and q and q* are as defined above. Note that since T,, may be expressed in closed form in terms of elementary functions, (3.12) yields the modes in terms of elementary functions only. From (3.11),

lim q

-mX

and

lim q*

w-aoo

=

m2 + v.

Thus,

lim w2f(W) = Constant.

w-foo

Hence the distribution of W can have no finite moments of order above zero.

Kappenman (1971) extended Press's ratio distribution for the multivariate case by considering the joint pdf of WT = (X2/X1, X3/X1, ..., XP/X1), where X = (X1i ... , XP)T is a p-variate t random vector with degrees of freedom v, mean vector µ, and correlation matrix R. The

3.3 Ratios

61

expressions for the joint pdf turned out to depend on whether p is odd or even. Introduce the following notation

VT =

(1, WT),

M = VTR-1V,

K = -2VTR-1µ, L = v + µT R-1 µ, _

a

L

K2

M

4M2'

and

K

b

2M Then the expressions for f (w) are a 1111

f (W)

_

-1/2

yv12l' ((v + p)/2)

(p-11)`/2

p-1

Irp/2 (Ma)(v+p)l2 r (v12) O G-1-2k ) a x

J _00

u2k {au2 + 11

2k

by -1-2k

-(,+p)/2

du

ifpisodd; f (w)

2abp-1 JR1-1/2 vvl2r ((v + p)/2) .7rp/2 (Ma)(v+p)l2 F (v/2) P/2

1: k=0

(:) p

2k-i

\b1

oo X

J

b/a

u2k-1 {a2u2 +

(p-2)/2

+

p-1

> (p2ki) \

a}-(v+p)l2

du

a) 2k

k=0

x

u2k {a2u2 + a} -(v+p)12 du


62

if p is even and b < 0; and f (W)

-

IR1-1/2 2abp-1 vv12r ((v + p)/2) 7rp/2M(v+p)/2P (v/2)

p/2

p_1

a 2k-1

F (P-) \b1 k=o X

r

Jb/a

u2k-1 {a 2u2 +a

(E-2)/2

+

k=O

p-1

I

du

(a)2k

(2k1) \bl 6/a

X

I - (v+p) /2

u2k {a2u2 +a) -(v+p)12 du

if p is even and b > 0. The integrals in these expressions can easily be rewritten in terms of the gamma and incomplete beta functions; see Section 3 of Kappenman (1971) for details.

4 Bivariate Generalizations and Related Distributions

In this chapter, we shall survey a number of specific bivariate distributions that contain Student's t components.

4.1 Owen's Noncentral Bivariate t Distribution Let Y1 and Y2 have the bivariate normal distribution with zero means, unit variances, and correlation 1. Let vS2 have the chi-squared distribution with degrees of freedom v and be independent of the X's. Then X 1 = (Y1 + 61) /S and X2 = (Y2 + 62) /S have the noncentral univariate t distributions with degrees of freedom v and noncentrality parameters 61 and 62, respectively. Owen (1965) studied the joint distribution of (X1, X2), a noncentral bivariate distribution. The marginal cdf of Xj, j = 1, 2 may be written as Pr (X j < _ y)

_ r(v/2)2(U-2)/2

\ fx1(x)(_8dx, Y_x

/

(4.1)

where 0 and 4 are, respectively, the pdf and the cdf of the standard normal distribution. Integrating by parts repeatedly, one obtains for odd values of v

Pr (Xj < y)

=

-D (-6jVB_) + 2T (6j IB, A)

+2 [M1+M3+ and for even values of v

Pr(Xj 3,

and the H's are defined recursively by

Ho = -c(R)4 (AR - S) and

Hk = ak+2RHk-1,

k > 1,

where

Lk-1

= ak+2RLk-2,

k>3

with the initial value L1 = (ABR/2)0(R)O(AR - 6) and

k>3

ak (k - 2)ak-1'

with the initial values a1 = a2 = 1.

4.2 Siddiqui's Noncentral Bivariate t Distribution Siddiqui (1967) considered the joint distribution of Student's t variates when (Y1i, Y2i), i = 1,.. . , N is a random sample from the bivariate normal distribution with zero means, unit variances, and correlation coefficient p. Let 1 N Y1

1'2

N

=N

Y1i,

i-1

N

Y2i, i=1

Si

=

1

N

N(Yii-Yi)2, i=1

S22

-

1

N

N

1: (Y2i - Y2)2 i=1

4.2 Siddiqui's Noncentral Bivariate t Distribution

67

and 1

R=

N

E(Y1i-YO (Y2i-Y2)

NS1S2 i=1

The interest is in the joint distribution of the Student's t variates

(X1, X2) =

Yi

Y2

( N-IS1' N-1S2

It is well known that the joint pdf of (Y1, Y2, S1, S2, R) is f (91, y2, S1, S2, r)

=

2 (N-4)/2

N N(802) N-2

(1 - r)

exp

27r2I'(N-2)(1-p2)N/2

+y2 + s2 - 2p (xy + rsi s2)

N 2(1-p2)

x2 + s21

}

for -oo < y1 < oo, -oo < 92 < 00, 0 < 81 < oo, 0 < s2 < oo, and -1 < r < 1 (see, for example, Kendall and Stuart, 1958). After suitable transformation, Siddiqui obtained the joint pdf of (X1, X2, R) in the form I' (v + 2) (1 - p2)

f (x1, x2, r)

=

(v+1)/2 (I

- r2) (v-2)/2

(27r)3//2/r (v + 3/2) (1 - b - cr) v+1l2

vl

X S 11+ x 2F1

I

(i+)} -(v+

1)/2

2

1 3 1+b+crl 2;v+2' 2 2,

C1

,

where v = N - 1, 2F1 is the Gauss hypergeometric function, b

_

Px1x2 V

1+

-+4 ,

and

c=

P

It is easily seen that the limit of this joint pdf as v -4 oo is trivariate normal with independent components (X1, X2) and R. Integrating out

Bivariate Generalizations and Related Distributions

68

r in (4.4), Siddiqui showed further that the asymptotic joint pdf of (Xi,X2) as v -i oo becomes r(v + 2) (1 - p2)("+1)/2

f (xl, x2)

-c2}1-'/2

{(1 -b)2

27r J F (v + 3/2) (1 + X2 1J

1

Sl

3

1-Jb2+c2

)

X 2F1 (21 2'v+2' 2(1-b)

The exact joint pdf of (X1, X2) was also given for the two special cases v = 1 and v = 3. For v = 1, the joint pdf reduces to a bivariate Cauchy distribution pl) { 1 + (2 B) cot B} , (4.5) f (xl, x2) -- 2 -(1+

-

x2) (1c+e 2)

where 2p (1 - p2) (1 + y1y2)

1+yi 1+y2

lx

In fact, if p = 0 in (4.5), then one arrives at a product of two independent Cauchy densities. For v = 3 the joint pdf is f (x1, x2)

2

-

p2) 351-2

(i+) (i+a) 2

2

2

j3,

where 13

r(9/2)r2(k + 1/2) = y., r(k+9/2)r2(1/2)k! J(1

1

1

2)

-05-21

- b - c)'-5/2 - (1 - b +

(k)

c)1-5/2 }

.

4.3 Patil and Liao's Noncentral Bivariate t Distribution Patil and Liao (1970) provided an extension of Siddiqui's work when (Yii, Y2i), i = 1, ... , N is a random sample from the bivariate normal distribution with zero means, common variance a2, and correlation coefficient p. Instead of considering the joint distribution of (4.3), they considered the joint distribution of

(X1, X2) _

SYI

,

SY2

,

(4.6)

4.4 Krishnan's Noncentral Bivariate t Distribution

69

where S is the pooled sample standard deviation defined by N

N

2

2

' - 1'1) + i=1 (Y 2i - Y2) .i.1 lYiz 2N - 1

S =

Exact expressions were given for the joint pdf and cdf of (X1, X2) and for the corresponding marginal distributions. For instance, if N is odd and equal to 2q + 3, then the joint pdf of (X1, X2) can be represented in the form f (X1, X2)

1

= 22q+sir(q + 1) (1

p2)q+3/2 r2(q + 1)

-

q

X

(q) k=O

(-1)q-k (.I_P

k

{ 2(1 + P) 1

1

p

2q-k 1=0

)

p

+ 2)

-1

2q-k+l

2

(1 -p2)

+

r(2q-k+1) y}_(k+2)

+

- 2Py1Y2

8(q + 1) (1 - p2)

F(k + l + 2) l!

1

2(1-p) -(k+1+2)

+ Y1

- 2Pyly2 + y2

8(q + 1)(1 - p2)

while if N is even, then the joint pdf of (X1i X2) becomes (1 + p)(1-N)/2 J (x1, x2)

_

,7r2N+l(N - 1)(1 - p) (N-1)12 X

k

((1 -N)12 ) k

1 - p2

}2 (N + k - 1) {

(

1

+ P) 2

- N +k

X

yi - 2Pyly2 + y2

1

{ 2(1 + p) + 4(q - 1)(1 - p2) }

4.4 Krishnan's Noncentral Bivariate t Distribution Krishnan (1972) provided another extension of Siddiqui's work when (Y1i, Y2i), i = 1, ... , N is a random sample from the bivariate normal distribution with means (-y, 8), unit variances, and correlation coefficient p. She considered the joint distribution of

=

(

N-1Y1 Sl

N-1Y2 S2

)'

(4.7))


70

where Sl and S2 are correlated chi-squared random variables independent from Yl and Y2, respectively. Series representations were derived for the joint pdf and the cdf of (X1, X2). One of the representations given for the joint pdf is

y

00

f (XI, X2)

= l

j=0

C1+N 11/

(1+N

21)

00

x E d,,,, m=0

where

A =

N 2 (1 - p2)'

A3NN24-N 3-N exp {-A (ry2 + j2 - 2p-y6) } 7rP ((N - 1)/2)

B

(2pAxlx2)2-k

yk Cj

(N -

1)j-k(j - k)!'

(pA) (N-3)/2+k / {(k/2)!P (N + k - 1)/2)),

Vk

0,

dm

,

if k is even, if k is odd,

(-2c)2(-2d)",-ZI,(N+i+j11,(N+j+m-al

= i=0

i!(m - i)!

2

C =

A (pa - 'Y) xl

d=

(p'Y -

J

2

J

N-1+xi

and S) X2

N-1+x2

In the central case -y = 8 = 0, (4.8) reduces to the form derived by Patil and Liao (1970).

4.5 Krishnan's Doubly Noncentral Bivariate t Distribution

71

4.5 Krishnan's Doubly Noncentral Bivariate t Distribution If Y is a normal random variable with mean S and unit variance, and S2 is an independent noncentral chi-squared random variable with degrees of freedom v and noncentrality parameter A, then

X =

Yv/-V-

S

is said to have the doubly noncentral univariate t distribution with degrees of freedom v and noncentrality parameters S and A. The properties of this distribution have been studied by several authors; see Robbins (1948), Patnaik (1955), Krishnan (1959), Krishnan (1967a), Bulgren and Amos (1968), and Krishnan (1968) - see also Chapter 31 in Johnson et al. (1995) for a summary. The pdf, the expectation, and the variance of X are given by 00

ao

E k=01=0

Ak exp {- (A + 52) /2} Fv2kk!17 (1/2 + 1) Be (1/2 + 1/2, v/2 + k)

2v) (1 + E(X) =

v-1

v S

2r (

2

1

)

1F1

-(v+1+2k+1)/2

x2

Sx

v

V

)

vl r \21'

a

(2' 2' 2)

v > 1,

and

Var(X) =

(1

V

+ S2)

v-2

1F1

(1, 2;

- 2) - {E(X)l2

,

v > 2,

where 1F1 denotes the confluent hypergeometric function. A bivariate analog of (4.9) was defined by Krishnan (1970) as follows. Let (Y1, Y2) follow a bivariate normal distribution with zero means, unit variances, and correlation coefficient p. Let (S1, S2) follow independently a noncentral bivariate chi-squared distribution with degrees of freedom v, noncentrality parameter A, and correlation coefficient p (Krishnan, 1967b). Then the random vector (X1,X2)

Y2-1,5

Sl ' (Yip

Sz

)

(4.10)

is said to have the doubly noncentral bivariate t distribution with degrees of freedom v and noncentrality parameter A. Krishnan (1970) derived

72


the corresponding joint pdf of (X1, X2) and provided an application involving the sample means and variances from two correlated nonhomogeneous normal populations. The special case of (4.10) for S1 = S2 = S was considered by Patil and Kovner (1969), who provided expressions

for the joint cdf of (X1i X2) and showed that when the means of Yj are zero the probabilities of (X1, X2) in rectangular regions are monotone functions of p. In the special case Si = S2 = S and A = 0, the distribution of (X1, X2) reduces to that of the central bivariate t.

4.6 Bulgren et al.'s Bivariate t Distribution Suppose Y1,. .. , Y,,,,, Y,,,,+1i ..., Ym+n denote iid normal random variables

with common mean p and common variance a2. Bulgren et al. (1974) considered the joint distribution of (X1, X2) defined by (X1,X2)

m+nY2

mYl

=

S2

n-1)S; m+n-2 )S2

(

where 1

Y1

m

=

i=m+1

mY1 + n,,*

Y2

S21

-

m+n

..,

E(Yi-Y1)2 1

i=1

and 1

S2

n-1

m+n

(Yi-Y*)2 i=m+1

The distribution of (X1, X2) is bivariate t with a different noncentrality

parameter for each variable. Note also that X1 and X2 have, respectively, m - 1 and m + n - 2 degrees of freedom. Bulgren et al. (1974)

4.7 Siotani's Noncentral Bivariate t Distribution

provided series representations for the joint pdf of (Xl, X2). central case µ = 0, 2 (m+n)/2 A

f (x1, x2)

xE1)3r(m+n+

(1+ 7

j!

\k)B

2

'

l2

\n(m-1))

2j-k

-_x2

xxi

n+m

/

2

)-

x2

m+n-2

)

\m+k n-1

2 (2j xk=o

In the

-(m+n)/2

2 2

Cl+m+n-2

j=0

73

(4.11)

n(m+n-2)

where

A=

2(m+n)/27f

n(m-1)(++ n-2)1, I m2 1)I'I n-2

1

In the noncentral case p $ 0, the joint pdf is `even more complicated. Letting n = am, a > 0, and taking m -+ oo in (4.11), one observes that the limiting distribution in the central case is the bivariate normal distribution with zero means, unit variances, and correlation 1/(1 + a).

4.7 Siotani's Noncentral Bivariate t Distribution Siotani (1976) considered the most general form of (4.6) introduced by Patil and Liao (1970). Let Y be a p-variate normal random vector

with mean vector p, unit variances, and correlation matrix R. Let S = (V2 +V2)/(2v), where (VI,V2) has the central bivariate chisquared distribution with degrees of freedom v and correlation coefficient T. Siotani derived the distribution of X = Y/S for general p and R. The

derivation required the joint pdf of (V1,V2) that was given by Siotani (1959) in the form cc

f (V1, V2)

=

v+2k2)(v+2k2/v+2k

1: Ck(T) k=O

1

(V1

(1-T )

2

x exp { l

r2 ((v + 2k)/2)

V1 + v2

2(1-T2)

,

where C k (T)

=

F (

1T

' X2)/2) (1 -

T2)v/2 r2 k.

(4.12)


74

From this, one can easily obtain the pdf of V1+V2 2v (1 - T2)

S

W=

T2

as

f (w)

Ck(T)f2v+4k(W), E k=0

_

(4.13)

00

where

=

f2v+4k(w)

2(2v) v+4k

2k)w2v+4k-1

2v+4kr(v +

exp {-vw2} .

(4.14)

Since cl(T) + + co. (T) = 1, (4.13) is a mixture of (4.14) with the weights given by (4.12). Thus the joint pdf of X is also obtained in the same form 00

Ck(T)f2v+4k (x)

,

k=0 where Ck(T) are given by (4.12) and

f+4k (x) = exp

{_,2TR,} r (v + 2k + p/2) (1 2

(2vx)p/2r (v + 2k) IRI112

x{1+

1 - T2 xTR-1x} - (v+2k+p/2) 2v

0 r (v + 2k + (p + l)/2) l!r (v + 2k + p/2) 1=0

L,

X

2 (1 - T2)xTR-1µ 2v + (1 - T2) xTR-1x

X

When p = 2, µ = 0, and p = T (p is the correlation coefficient between Y1 and Y2) this coincides with the pdfs derived by Patil and Liao (1970).

4.8 Tiku and Kambo's Bivariate t Distribution Suppose (X1, X2) has the bivariate normal distribution with means (Al, µ2), variances (o, , a2), and correlation coefficient p. Its joint pdf can be factorized as f (x1, x2)

= f (x1

I x2) f (x2)

4.8 Tiku and Karnbo's Bivariate t Distribution

75

where

f(xiIx2) =

1

Ql

1 - p2

expl-

1

z z 2Q1 (1 - p)

al

i

x1-µl

z

(4.15)

20'2 (x2 - µz)2

(4.16)

-p-92(x2 - µ2) and

f (x2)

1

a U2

exp S l

z

Numerous nonnormal distributions can be generated by replacing either f (xl I x2) and/or P X2) by nonnormal distributions. Tiku and Kambo (1992) studied the family of symmetric bivariate distributions obtained by replacing (4.16) by the Student's t density f (x2)

a

(x2 -

µ2)2

1+

V

(4.17) kor2

where k = 2v - 3 and v > 2. This is motivated by the fact that in many applications it is reasonable to assume that the difference Y1 - µl P(91/92)(Y2 - µ2) is normally distributed and the regression of 11 on Y2 is linear. Moreover, in numerous applications Y2 represents time-tofailure with a distribution (Tiku and Gill, 1989; Gill et al., 1990), which might be symmetric but is not normally distributed. Besides, most types of time-to-failure data are such that a transformation cannot be performed to impose normality on the underlying distribution (see, for example, Mann, 1982, page 262). On replacing (4.16) by (4.17), the joint pdf of (Xl, X2) becomes }_V

f (XI, X2)

=

1

Qloz

{i+ (X2 - µ2)2 ka2

k(1 - p2)

x exp -

1

2Q1

1

x1 - µl -P - (x2 -µ'2)2

(1 - p2)

z

1J

(4.18)

Limiting v -+ oo, (4.18) reduces to the bivariate normal pdf. Writing p2j = E((Y1 - µ1)2(Y2 - µ2)j) for the cross product moment of order i + j, one observes that all odd-order moments are zero and that the first few even-order moments are 2

76

Bivariate Generalizations and Related Distributions P11

= P6192,

/

= 3al I

µ4o

=

P31

P22

/43

=

3pal

2P4

1 + 2v

5

I

22 (i+ 2v-5

a2

P

)

I

6 P2

cic2 (1+2 p2 +

= 3pala23

(1 +

2v - 5 2

2v '5

'

and µ04

=

3(2v - 3) 2v - 5

4

a2

In fact, the moment generating function (mgf) of (Y1,Y2) is given by E [exp (01X1 + 02X2)]

= exp I Al - pal µ2 101 +

(1

81 12

2

OF2

xM2 02 +Cpl a µ201J

_2

,

92

denotes the moment generating function of X2. This moment generating function does not exist unless, of course, v = oo. However, the characteristic function does exist and is given by Sutradhar (1986). Estimation issues of the distribution (4.18) are discussed in Section 10.1. where

4.9 Conditionally Specified Bivariate t Distribution Let (X, Y) be a continuous random vector with joint pdf fx,y (x, y) over R2. Let fx(x), fy(y) and fxI y(x I y), fy, x(y I x) denote the associated marginal and conditional densities, respectively. Assume that X I Y and Y I X are Student's t-distributed with the pdfs

fxIr(x I y) =

1'((v + 1)/2) V"7-1'(v/2)

V/0(y) {1 +

or(y)x2}-(-+1)/2 (4.19)

4.9 Conditionally Specified Bivariate t Distribution

77

and

r(r(v/2)

fyix(y I x) =

+r(x)y2}-(v+1)/2

T(x)

(4.20)

where x E 2, y E Q, v > 0, and a(y), r(x) are some positive functions. Writing the joint pdf of (X, Y) as a product of marginal and conditional densities in both possible manners, one obtains

fr(y)

a(y) {1 + Q(y)x2}-(u+1)l2

= fx(x) r(x) {1 +T(x)y2

(4.21)

where x E R and y E R. Set 2/(v+1)

2/(v+1)

g(y) _{fr(y)

h(x) _ {fx(x) T(x)}

ma(y)}

(4.22)

so that, after rearranging, (4.21) becomes g(y)

+ y2g(y)T(x) - h(x) - x2h(x)o, (y)

=

0,

(4.23)

which must be solved for a, T, g, and h. Kottas et al. (1999) recognized that (4.23) is a special case of the functional equation n E fk(x)9k(y)

=

0,

k=1

whose most general solution is given in the classical book by Aczel (1966,

page 161). Thus, with h(x), x2h(x) and g(y), y2g(y) being the systems of mutually independent functions, the solution of (4.23) is found to be T(x)

_

_

_3 + _4x2 Al + A2x2'

a(y) =

A2 + A4y2

(4.24)

Al + A3y2

and (x)

h

1

1

=

Al + A2 X2'

9(y) =

(4.25)

Al + A4y2

for A3 E R, j = 1, 2, 3, 4. Finally, substituting (4.22), (4.24), and (4.25) into (4.21), the joint pdf is derived as

fx,y(x,y) = Nv(A1, A2, A3, A4) x {A1 + A2 X2 + A3 Y2 +

A4x2y2}-(v+1)12

, (4.26)

denotes the normalizing constant. Utilizing certain compatibility conditions given in Arnold and Press (1989), Kottas et al. found that (4.26) is a well-defined joint pdf if Al E R+U{0}

where x E 1, y E l1, and

78


and Aj E R+, j = 2,3,4. Moreover, if Al = 0, then one must have v E (0,1). The normalizing constant is given by the integral 1

Nv(A1,A2,A3,A4) 00 A4x2y2)-(v+l)/2

(A1 + A2x2 + A3 Y2 +

J7.

dxdy. (4.27)

00

In the case Al 54 0, making the transformation s = (A2/A1)x2, t = (A3/Ai)y2, letting 0 = A1A4/(A2A3), and using the integral representation of the Beta function, B(a, b)

=

J0

a> 0,

o" xa-1(1 + x)-a-bdx,

b> 0,

one obtains

_

1

Nv(A1, A2, A3, A4)

B (2'

dx

A(v-1) /2 2)12A3

(1 + x)v/2

0

x(1 +

Ox)

Letting w = x/(1 + x) and manipulating, Kottas et al. obtained v

Nv(A1, A2, A3, A4)

=

2

B (21, 2L) 2)

1

(1

2 31 /1

12' 2'

(4.28)

2

where 1

I (a, b, c; x)

=

L

for c > b > 0. In the case Al = 0, similar arguments show that

N(0

,

A 2, A 3, A 4 )

(A2A3)v/2A41-v)/2 _ B(1 vB(1=v v,

2'2

2

'21

where 0 < v < 1. The integral (4.29) converges for z < 1. For I z I> 1, Kottas et al. provided an alternative representation of (4.28) in terms of the Gauss hypergeometric function (see, for example, Magnus et al., 1966, page 54). It is also possible to represent (4.28) in terms of elliptical

integrals of the first and second kind (Carlson, 1977, Chapter 9). For example, if v = 1, then (4.28) can be easily rearranged to yield Nv(A1,A2,A3,A4)

=

V1 A4

27rRF(0,1/0,1)'

4.9 Conditionally Specified Bivariate t Distribution

79

where RF is the elliptical integral of the first kind defined by RF(a, b, c)

=

2

f00

{(x + a)(x + b)(x + c)}-1/2 dx

J0

with a, b, c nonnegative, and at most one of them equal to zero.

If 0 < v < 1, then (4.26) does not possess finite moments; thus, from here on we shall consider the case v > 1. If v > max(m, n), for non-negative integers m, n, then Kottas et al. showed that

E (X '"Y"`

=

r(

)2

)r(v m2

)I(

2

1 1 , V+1; [11"[12 r (L) V) I (1, 2 2 2

;1-0) -(4-30)

- 0)

provided that both m and n are even or zero. The expectation is zero if at least one of m or n is odd. This suggests that the distribution may be an appropriate model for uncorrelated but nonindependent data. From relations (4.21), (4.24), and (4.26) it is immediate that the marginal densities are

fx(x) =

11

{I

(1,

11

v 22,

2

- 0)1

+µix2)v/2}-

1 +Oµ1x2(1

and fY(y)

=

1 µ2{I 22 v21;0) 1+Oµ2y2(1+µ2y2)vl2}

where x E l and y E R. Here, µj _ Aj+1/A1, j = 1,2 are the intensity

parameters while ¢ and v are the dependence and scale parameters, respectively. It is easily noted that X and Y are independent if and only if 0 = 1. The graph of the joint pdf is symmetric and bell-shaped and takes the standard form when µ1 = µ2 = 1. From relations (4.19)-(4.20) and (4.24)-(4.25) it is immediate that X I Y has the Student's t distribution with degrees of freedom v and scale parameter (1/v1)(1 + µ2y2)/(1 + 0µ2y2), and that Y I X is also Student's t with degrees of freedom v and scale parameter (1/v2)(1 + µ1i2)/(1 + Oµ1x2), where µi = .j+1/)1, j = 1, 2. Consequently, the conditional moments are

E(Xm I Y=y) =

r (m+l) r (v2

2

V'571' (2)

1 +µ2y2

m/2

µ1 (l + Oµ2y2) /

and

E(Y" I X = x) =

r(I)r(v=m) 2

V/7rr(2)

l+[11x2 2 µ2(1 +Oµ1x)l

m/2

80


provided that m is an even number less than v. If m is odd, then the corresponding conditional moments are zero. In the special case v = 1, (4.26) reduces to the centered Cauchy conditionals model of Anderson and Arnold (1991). The limiting case 0 -4 0 gives the bivariate Pearson type VII distribution (Johnson, 1987, page 117) with location parameters equal to zero and uncorrelated components. If 2v is a positive integer, then this limit distribution reduces to a special case of the general bivariate t distribution (see, for exam-

ple, Johnson and Kotz, 1972, page 134, relation 1) with uncorrelated components and 2v degrees of freedom. For p, = µ2 and v = 2, the limit distribution reduces to the bivariate Cauchy distribution (Mardia, 1970a, page 86) while for µl = 42 and v = v + 1 it gives the bivariate t distribution (Johnson and Kotz, 1972, page 134, relation 2) with v degrees of freedom. In the latter case, the standard bivariate normal distribution with independent components arises as a further limiting case when v -+ oo. Other special cases of (4.26) are the centered normal conditionals model studied by Sarabia (1995) and the Beta conditionals model of the second kind (Castillo and Sarabia, 1990).

4.10 Jones' Bivariate t Distribution Let Z1, Z2, W be mutually independent random variables with Zi having the standard normal distribution and W having the chi-squared dis-

tribution with degrees of freedom n1. Then the standard bivariate t distribution with degrees of freedom n1 is the joint distribution of n1Z2

Vn1Z1 W_

I

VV_

)

(4.31) *

One disadvantage of this model is that the two univariate marginals (which are Student's t) have the same degrees of freedom parameter and hence the same amount of tailweight. Jones (2002b) provided an alternative distribution with Student's t marginals, each with its own arbitrary degrees of freedom parameter. Precisely, if W1, W2 are independent chi-squared random variables (also independent of Z1, Z2) with degrees of freedom v1 and v2 - v1, respectively, then Jones (2002b) considered the joint distribution of (X1, X2)

=

v1Z1

v2

Z2

(4.32)

VIWI ' V' Wl +W2 Note that the ith marginal of this distribution is Student's t with degrees of freedom vi. It is easy to see that the correlation between X1 and X2

4.10 Jones' Bivariate t Distribution

81

is zero, a property also shared by (4.31). If rl < vl and rl + r2 < v2, then the product moment is given by

E(Xi'X22) _

vii/2v22/2r(l+ri)r(l+rz)r(vi-rl\r/v2-rz) 2 2

2

-2

J

7rr(2)r( 2

1

if rl and r2 are even and is zero otherwise. The joint pdf of Xl and X2 is

f (xl, x2)

x2 \ -(1+v2/2)

x2

CI1+vi+1121 1 + 112 . X 2 v

v2

v2 -

2F1 1+ 2

2

2

Cl +

`

x12 V1

y1

2

+

2 112

(4.33)

/

where C

r (1+V1 ) r (

1

-

7f

V217 2

2

and 2F1 denotes the Gauss hypergeometric function. The conditional pdf of X2 given Xl = xl is x2) -(1+L 2/2)

f (x2 I x1)

C

(ul + 21 112

/

X 2F1 11+

ul-1

112 V2-v1 1+112

\\\

2'

2

'ul+x2/ 2V2)

2

(4.34)

where ul = 1 + x 21v, and

C =

u(l+vl)/2r (1 + 7fv2

(l+v \z 2

)

If v2 + 1 > r, then the conditional rth moment is given by E

(Xr

2

I X, = x l) = 1

yr/2u(r-V2+vl)/2 2 1

r(

r ( 1+v2-r ) \ "r( ) 2

)

2

1+112 2

X 2 F1

C

v2-r+1 112-vl 1+112 U1 2

2

2

'

1)

U1

(4.35)

if r is even and is zero otherwise. Setting 112 = vl = v in (4.34)(4.35), one obtains the corresponding forms for the standard bivariate t distribution (see Section 1.11). Note that the conditional variance


82

Var(X2 I X1 = x1) increases with I x1 I. In a parallel fashion, with u2 = 1 + x2/v2, the conditional distribution of X1 given X2 = x2 is

f (xl

I X2)

= C

-(1+vi/2)

x2

u2 +

V11

U2 V2-v1 1+1/2

X 2F1 1+

2'

2

2

U2 1

u2 + X211V1

where

C=

u(1+v2)/2r

(1) r' (2) r (1 + 2 ) Vrvlr(z)r(1 )

This time, the conditional rth moments exist provided v1 > r, unless v1 = v2, in which case one needs 1 + v1 > r. The odd conditional moments are again zero and the even conditional moments are given by the simpler form r E' (X1 I X2 = x2)

r/2u2r/2r (12r) r (v) r (2) r

=

v1

(v2

2+1)

r (2) r (1±v2) r () 2

2

The construction (4.32) can be easily extended to the multivariate case. Two straightforward extensions are

Let Z1,.. . , ZP, W1, ... , WP be mutually independent random variables with Zi having the standard normal distribution and Wi having the chi-squared distribution with degrees of freedom vi - vi-1. Then, (Xi, X2, ... , Xp)

-

Z1

v2 Z2

v1

W1

+W2'...,

' 11W1

vP Zp W1 +...+WP

(4.36)

has a multivariate distribution with univariate marginals that are Student's t distributed with degrees of freedom vi, i = 1,.. . , p. The bivariate marginals of (4.36) have the distribution of (4.32). With the notation as above,

(X 1,XZ,...,X'p)

-

v1 Z1 W1

v2 Z2

W1 + U2 '

... '

vP ZP

W1 + Up

(4.37)

has a multivariate distribution with the same univariate and bivariate marginals. Here, Ui are independent chi-squared random variables

(also independent of Z,, Wi) with degrees of freedom vi - v1, i = 1,...,p.


(b)

(a)

-2

0

83

1

2

-2

0

1

2

X1

X1

Fig. 4.1. Jones' bivariate skew t pdf (4.33) for (a) vi = 2 and v2 = 3; and (b)

v1=2andv2=20

Further extensions of (4.32) arise by adding further independent chisquared random variables inside the square roots in the denominators of all variables in (4.36) or by adding a single further independent chisquared random variable inside the square root in the denominator of Xl in (4.37). Jones (2002a) provided another bivariate generalization of (4.31). This generalization has the skew t distribution (Jones, 2001a) as its marginals. If U denotes a standard beta random variable with parameters a and c, then a skew t variate is defined by

X-

a + c(2U - 1) 2 U(1 - U)

84


The corresponding pdf is

f(x;a,c)

-

21-a-c17(a+ c)

x

) )1+

a -+c I(a)I(c)

l a±II z

a + c + x2

}

}C+i/2

r

x{ 1+

(4.38)

a+c+x2

111

The standard Student's t is the particular case for v = 2a when a = c. If a > c, then (4.38) has positive skewness; also, f (x; b, a) = f (-x; a, b). Further details about (4.38) are given in Jones and Faddy (2002). The bivariate generalization proposed in Jones (2002a) is constructed in the same way as (4.38): Specifically, if (U,V) denotes a Dirichlet random vector with the joint pdf f(u,v)

=

r(a + b + c) ua-lvb-1(1

r(a)F(b)r(c)

- u -

v)c-1

(where u > 0, v > 0, and u + v < 1), then define (X1,X2)

-

(4.39)

1 - U ' 2 U1 12U

2

U where d = a + b + c. It can be verified that the joint pdf of (X1, X2) is

f (xl x2) =

dF(d + 1)

2d-1r(a)r(b)r(c)

a-1

xl

1+

d+x1 b-1

1

x

1

X2

V

d + x2 2

x2

d+x2

-

1

(d + x1)3/2 (d + x2)3/2

xl

d+xl

C

(4.40)

Because of a direct analogy with the Dirichlet distribution, only one of the two marginals of (4.40) can be a symmetric Student's t distribution, the other necessarily being skewed. This Student's t distribution will have degrees of freedom d, and any skew t marginal will have a total

parameter value of d, but divided up into unequal amounts. In this sense, marginals of (4.40) are most closely associated with Student's t distributions with degrees of freedom d. Note that if instead of (X1, X2) in (4.39) the transformation was made to (-X1, X2), then one would have obtained the equivalent distribution on x1 + x2 > 0. Also, (-X1, -X2) would have given the equivalent

distribution on x2 < x1 and (X1, -X2) as the same on x2 + x1 < 0.


85

(b)

(a)

Nx

0

8

X1

X1

(C)

(d)

12

N

(0

N

N Nx

4

NX

N

X1

xt

Fig. 4.2. Jones' bivariate skew t pdf (4.40) for (a) a = 1, b = 1, and c = 1; (b)

a=3,b=4,andc=5;(c)a=5,b=1,andc=1;and(d)a=1,b=5, and c=1

The corresponding changes to (4.40) would simply have been to make corresponding changes to the signs of x1 and x2. The means and variances associated with (4.40) can be easily obtained from the results provided in Jones (2001a)

E(X

1

)

_

vfd-

(a

(b + c - 1/2) -b -c)r (a-1/2)r r(a)r(b + c)

(a

- b - c)

2

E(X2 ) =

Vd2

r (a + c - 1/2) r (b-1/2) r(a + c)r(b)

Var(XI) = d(a-b-c)2+d-2 _{E(X1)}2,

4 (a-1)(b+c-1)

86


and Var (X2)

=

d (a - b + c)2 + d - 2

4 (a + c - 1)(b - 1)

_ {E (X2)}2

.

The covariance between X1 and X2 appears not to be available in closed form.

5

Multivariate Generalizations and Related Distributions

This chapter contains a large number of modifications and extensions of the standard multivariate t distribution introduced in (1.1). Some of them are of somewhat complex nature. It thus requires a careful reading to see the forest behind the trees!

5.1 Kshirsagar's Noncentral Multivariate t Distribution one of the earliest results in the area of noncentral multivariate t distributions is that due to Kshirsagar (1961). Let Y be a p-variate random vector having the normal distribution with mean vector µ, common variance a2, and correlation matrix R. Let S2 be distributed independently of Y according to a chi-squared distribution with degrees of freedom v. Kshirsagar (1961) considered the distribution of X = Y/S and showed that it has the joint pdf .f (x)

= exp {

l

r ((v +p)/2) J (vx)pl2r(v/2) JR1 "2

2

l

(

x{1+1xTR-1x} ll

00

E where

v

(-+p)/2

JJJ

r ((v + p + k) /2) k!r ((v + p)/2)

v/2_xTR-1

v + xTR-lx

k ,

(5.1)

= µ/a. This noncentral distribution reduces to the form of

(1.1) when µ = 0. We noted earlier in Section 1.11 that, if X has the central multivariate t distribution with degrees of freedom v and correlation matrix R, and 87

88


if

X= and

( R11

R

R21

R12 1

R22 ) '

where X1 is p1 x 1 and R11 is pi x p1, then

Z1 = x1 and

Zz =

v+p1

1/2

11+ X1 R111X1 J

`

V

(x2 -R21R111X1)

v

are independently distributed, each according to a central multivariate t distribution. This result does not remain true to the noncentral distribution (5.1). Actually, Siotani (1976) showed that if

is the partition of l; corresponding to that ofCCX, 52.1 = b2 - R21R111S1 and

R22.1 = R22 - R21R1i R12'

Z2 is then the joint pdf of Z1 and(_TRi1'1 .f (Zl,Z2)

= Kexp

C

tt - 1'ttT 1Rz21SZ.1

xtv (zi; R11,pi) tv+p, (Z2; R22.1,p - p1)'

where the last two terms denote the pdfs of central multivariate t distributions with appropriate parameters and K is given by the formidable expression

K = r 00 IF ((v+p+k+l)/2) (2)k/2 ( 2 1/2 FF- k!l!F((v+p)/2) v v+p-p1 k=O c=o k

ZTR-% 11

I

5.1 Kshirsagar's Noncentral Multivariate t Distribution

(z R22

X

t

89

152.1)l

pl)}(k+l)/2

tt {1 + z2R22 1S2/(v +

Siotani (1976) also derived the corresponding noncentral distribution when X is partitioned into k sets of variates as in (1.20). Following the notations defined by (1.21), (1.22), (1.23), (1.24), and (1.25), the joint pdf of

= X1

Z1

and

=

Z1+1

v

+

ql

I 1 + 1 XTR1 1X(1)1

- 1/2

- R(II 1)TR-IX(l))

(x(1+1)

is given by the lengthy expression

f (Z1,...,Zk)

exp

(o? I

2 1=1

(v + 41-1) r ((v + ql-1)/2)

11=1

-(-+ql)/2

-1 ZIT

k

1+

X

r((v+gl)I2) IRII.(1-1)I-1/2

k

k

v+ql-1

l=1 00

00

11=o

Ik=0

r((v+p+l1+...+lk)/2)

X

T, ((v +p)/2) t 11...1 k )Im/2

X

k

m= 1 k

2 ( v + qm-1

z1 R 1

tt

m

X

M=I (1 + ZrRmm.(m-1)zm) where

_m - R(m) (m 1) TR-1 (m-1) (m-1) and 62

-

Sr(m-1)Rmm(m-1) m(m-1)

1m

1


90

5.2 Miller's Noncentral Multivariate t Distribution Let Y have the p-variate normal distribution with mean vector µ and correlation matrix R > 0. Let S be distributed independently of Y according to a v-variate normal distribution with mean vector A and correlation matrix mIp. Miller (1968) considered the joint distribution of

XT =(XI,X2,...,XP)= C1Si1,l52I,...,ISPI I, which he referred to as the generalized p-dimensional t random vector. Assuming I S 12 has the chi-squared distribution, Miller showed that the joint pdf of X is given by 21-(v+p)/2m-v/2F(v

f (x)

_

r(v/2)-P/2

+p) (!+XTR_1X) m

JRI1/2

T _1 x exp -1 µR µ }exp

xD_(v+p)

-(v+p)/2

m (xTR lµ)2 4 (mxTR-ix + 1) }

xTR-1 µ mxTR-ix + 1

(5.2)

'

where D_(v+p)(.) is the parabolic cylinder function (see, for example, Erdelyi et al., 1953). If µ = 0, then (5.2) reduces to

f (x)

=

+ p)/2) (m F(v/2)irP/2IRI1/2 1 + xTR m vl2r((v x exp

2m

iFi

v

-

(v+p)12

1x

+P V 2

IAIz

' 2' 2m (mxTR-1x + 1)

where 1F1 is the confluent hypergeometric function (see, for example, Erdelyi et al., 1953). If both µ = 0 and A = 0, then (5.2) reduces

to the usual central multivariate t distribution (1.1) with degrees of freedom v and correlation matrix R. To the best of our knowledge, this interesting distribution given by (5.2) has not been pursued further since its introduction some 35 years ago.

5.3 Stepwise Multivariate t Distribution Let Y be a p-variate normal random vector with mean vector µ, common variance a2, and correlation matrix R > 0. Let vS2 /a2 be a chi-squared

5.3 Stepwise Multivariate t Distribution

91

random variable with degrees of freedom v, distributed independently of Y. Then the joint distribution of Y1

X1 =

S

and Yk

Xk

72

r(1_V

-r(k_1)Rk11Y(k -1) S

J1+i_YT S2 k_1)Rk11Y(k-1), k = 2,...,p,

(5.3)

where rk denotes the multiple correlation coefficient between Yk and (Y1,...,Yk-1), Y(k)

R(k)

=

(Y1,Y2,...,Yk), 1

r12

"'

rkk

r21

1

...

r2k

rkl

rk2

_

"

1

and

r(k)

=

(rl,k+1, r2,k+1, ... i rk,k+1)

is known as the stepwise multivariate t distribution. This distribution has applications in linear multiple regression analysis; for instance, suppose that a random sample Y 1 ,. .. , Y,,,, corresponding to some nonrandom values (zii, z2i), i = 1, ... , n, is available. The null hypothesis to be tested is that the slopes of the two simple regression lines, Y on z1 and Y on z2, are both zero. Then, the X1 and X2 above could correspond to the usual t statistics for testing the two regression coefficients (Steffens, 1974). Steffens (1969a) studied the distribution of (5.3) for the special case R = Ir, the p x p identity matrix. In this case, since 1k = 0, r(k_1) = 0 and R(k_1) = I(k_1), (5.3) reduces to

Xk =_

/v+k_1Yk//1yTy

(k_1 )

92

Multivariate Generalizations and Related Distributions k-1

Vv+k-1S

2

Y S

+

Yk

v+k - lYk vS2 + Xl +

+ Xk_1

If µ = 0, Steffens (1969a) showed that X1, X2, ... , XP are independent Student's t random variables with degrees of freedom v, v+1.... , v+p-1, respectively. This result also holds for general R (Siotani, 1976, Corollary 3.1). In the noncentral case µ # 0, the XD's are still independent, but X1 has the noncentral distribution with degrees of freedom v and noncentrality parameter ul /a while the XD's (j = 2,3,.. . , p) have the doubly noncentral t distributions with degrees of freedom v + j - 1 and noncentrality parameter p, /u in the numerator and (µ1 /Q)2 + (p2/0,)2 + + (µi_1 /U)2 in the denominator. Steffens derived the joint pdf of the X3 .'s in the bivariate case as the double infinite series f (X1, X2)

=

exp{- (62 +62) /2} v(v + 1)F (v/2)

oo

00

1

(2

xk k!

V+1

l!

vI (v 62)k

(v + 1)1/(2k)

x2 ( I+l

(v+k+1+2)/2

+ x22

x2 x-

(y251)l

v1/(21)

k=O 1=0

v xI'+2k+21+1/ 1

-

-(v+k+1)/2

where 6j = µa /v, j = 1, 2. For general p and R, Siotani (1976) showed that if µ = 0, then X1, X2, ... , XP are still independent Student's t random variables with degrees of freedom v, v + 1, ... , v +p - 1, respectively.

In the noncentral case, the joint pdf of the Xj in (5.3) generalizes to f (xl,...,xp)

_

1

P

F ((v + k)/2)

P

2

2kTk 11 v+k-1F((v+k-1)/2) k=1 k=1

exp

P

x

x2

k 1+v+k-1

-(v+k)/2

1=1 CO

°O

x E ... k1=0 P

X

l=1

kp=0

F((v+k1 ++kp)/2) F ((v + p)/2)

2

v+l-1

k,/2

kl!...,p.

D

5.4 Siotani's Noncentral Multivariate t Distribution

1+x

93

v+l-1

where the rk's are the noncentrality parameters given by Tk

-

k

- r(k-1)R(kl 1)S(k-1) r 2k

and i;j = pj/o-.

5.4 Siotani's Noncentral Multivariate t Distribution In Section 4.5, we discussed a bivariate generalization of the doubly noncentral univariate t distribution given by (4.9). Siotani (1976) provided a multivariate generalization of (4.9) by observing that the pdf of S* = S// (where S is a noncentral chi-squared random variable with degrees of freedom v and noncentrality parameter A) can be expressed as the Poisson mixture CO

f (s*)

=

1: Pk(A)fv+2k (S*) k=0

where Pk(A)

_

exp(-A)Ak k!

is the kth probability of the Poisson distribution with parameter A and fv+2k (S*)

2v ("+2k)/2

=

(S*)v+2k-1

2(v+2k)/2f ((v + 2k)/2)

1

exp

ll C-2vS*2J

0

He defined X = Y/S* to have the doubly noncentral multivariate t distribution, where Y is a multivariate normal random vector with mean

vector µ, unit variances, and correlation matrix R. The joint pdf of X is easily obtained as a Poisson mixture of the noncentral pdf (5.1) with v + 2k in place of v in the arguments of gamma functions and in the power of 1 + xTR-lx/v, that is, P X)

=

0" k=0

Pk (A) fv+2k (x) ,

(5.4)

0


94

where

I ((v+2k+p)/2) fv+2k(x)

l

T

2

(v7r)rl2r ((v + 2k)/2) IRI1/2

l

r

(v+2k+p)/2

x S I+ ixTR-1x } v

11

)))

k

Vf2-XTR-1

°O ['((v + 2k + p + l)/2) x l!F ((v + 2k + p)/2)

v + xTR-lx

5.5 Arellano-Valle and Bolfarine's Generalized t Distribution Arellano-Valle and Bolfarine (1995) considered what is being referred to as a generalized t distribution within the class of elliptical distributions. The distribution is defined by

X = µ + V1/2Y,

(5.5)

where V has the inverse gamma distribution given by the pdf (v/2)+1

v/2

f(v) _

\v)

(r(vl2)

(_A exp

2

)

'

v>0

and Y is distributed independently of V according to a p-dimensional normal distribution with mean vector 0 and covariance matrix R. We shall write X - tp(µ, R; A, v). When A = v, this distribution reduces to the usual multivariate t distribution (1.1) with mean vector it, correlation matrix R, and degrees of freedom v. For R > 0, the joint pdf of

X -tp(µ,R;A,v) is f (x)

_

[1+(X_)TR1(X_)]

F ((v + p)l2)

(v+p)l2

RI12

E (X) = µ,

v>1

(5.7)

and

Var (X)

=

v

A

2R,

v > 2.

Furthermore, for an m x 1 vector q and an m x p matrix B,

Z = 77+BX

(5.8)

5.5 Arellano-Valle and Bolfarine's Generalized t Distribution

(77+Bµ)+V1/2BY t,,, (i + Bµ, BRBT; ,\ v)

95

(5.9)

since BY has the m-dimensional normal distribution with mean vector 0 and correlation matrix BRBT. Now let

X=

/ 1

X2

I

(5.10)

,

(5.11) and

R =

R11

R12

R21

R22

(5.12)

where Xl is m x 1, R11 is m x m and so on. Taking B = [I,,,,, 0] in (5.9),

note that Xl - tn,,(pl, R, 1; A, v). By symmetry, X2 - tp_m(µ2i R22; A, v). Assuming R > 0, let = µl + R12R22 (x2 - A2),

Al (x2)

R11 - R12R22 R21,

R112 and

q (x2)

=

(x2 - µ2)T R22 (x2 - A2)

Using the fact that IRI

=

and

(x - µ)T R-1 (x - µ) _ (xl - µ1)T Rl11.2 (x2 - µ1 (x2)) + q (x2) note that the conditional pdf of Xl given X2 = x2 is given by

f (xl

I X2)

_

,

)}("+p-m)/2 F ((v + p)/2) {A + q (x2 A + q (x2) 7rm/2f ((v + p - m)/2) IRI1/2

-(v+p)/2

+ (x1 - Al

(X2))T

R11'.2 (xl

- µl (x2))

This means that X1 I X2 =x2 ^'

tm(µ1 (x2),R112;A+q(x2),v+p-m). (5.13)

96


Note that when.\= v, Xi I X2 = x2 - tm(µi (x2) , R11.2; v+q (x2) , v+ p - m). Since q(x2) 34 p - m, this shows that the usual t distribution does not retain its conditional distributions (see Section 1.11). Finally, it follows from (5.13) and (5.7)-(5.8) that

=

E (X1 I X2)

Jz1

+ R11R22 (X2 - 1i2)

and

0027 (X 1 I X 2 )

_

+ (X2 -µ2)T R22 (X2 - A2)

v+p-m-2

x (R11 - R12R22 R21) .

Arellano-Valle and Bolfarine (1995) also presented characterizations of the generalized t distribution (5.5) in terms of marginal distributions, conditional distributions, quadratic forms, and within the class of compound normal distributions. Briefly, these characterizations are Let X have the p-variate elliptically symmetric distribution with mean vector It and covariance matrix R (for a definition of an elliptically symmetric distribution see, for example, Fang et al., 1990). Then, any marginal distribution is a generalized t distribution if and only if X has a generalized t distribution. Let X = (XT,, XZ )T have the p-variate elliptically symmetric distri-

bution with mean vector µ and covariance matrix R, where X1 is m x 1. Then, the conditional distribution of X1 given X2 is the generalized m-variate t distribution if and only if the distribution of X is the generalized p-variate t distribution. The proof of this result, which assumes the existence of a density, is similar to the proof considered in the pioneering paper by Kelker (1970) for the characterization of the multivariate normal distribution. Let X have the p-variate elliptically symmetric distribution with mean vector 0 and covariance matrix Ip, and let A be a symmetric p x p matrix. Then, XT AX - (mA/v)F,,,,,u if and only if X - tp(0, Ip; A, v),

A2 = A, and rank(A) = in. This result is proved by utilizing Anderson and Fang's (1987) assertion on the spherical distributions that put zero mass in the origin. The fourth characterization within the class of compound normal distributions is a consequence of a well known theorem due to Diaconis and Ylvisaker (1979), which asserts that, in the regular exponential family

with the natural parameterization, if the posterior expectation is linear, then the prior distribution must be conjugated. It states that if

5.6 Fang et al.'s Asymmetric Multivariate t Distribution

97

X1, X2, ... is an infinite sequence of orthogonally invariant random vari-

ables (which means that for each p, X = (X1,. .. , XP)T and I'X are identically distributed, for all p x p orthogonal matrices r) such that X1 = 0 with probability zero and Var(X2 I X1)

= b + aXi,

0 < a < 1,

b > 0,

(5.14)

then X is distributed as tp(0, Ip, b/a, (a+1)/a). The converse also holds. Arellano-Valle et al. (1994) pointed out that (5.14) could be extended to yield a location mixture of generalized t distributions as follows. Let X1, X2, ... be an infinite sequence random variables such that for each p, X = (X1, ... , Xp)T and FX are identically distributed, for all p x p orthogonal matrices I' satisfying rip = ip (where lp is a p-dimensional vector of 1's). Under this assumption there exists random variables M and V such that, conditional on M and V, X1i X2, ... are independent and normally distributed with mean M and variance V. Actually, M and V can be interpreted as the limits n 1

Xn = n

Xi

-4 M

i=1

and

(Xi-Xn)2 sn= 1n i-1

-- V

as n --* oo, where the convergence is with probability 1. Furthermore, if

Var {(X2 - M)2 I X1, M} = a (Xi - M)2 + b,

0 2,

and (

Cov X

)

=

v

r

(v-2)(v-4)E[1 v>4,

2(v + 4)aaT E

ir(v-2)(1+aTEa)

1

'

5.7 Gupta's Skewed Multivariate t Distribution

101

respectively. Furthermore, using the definition (5.18), the product moments are easily obtained as µrl,r2i...,rp

=E

11

j=1 P

vr/2E (W-r) E II Yjr' j=1

_

v r ((v/2) - r) E r(v12)

2

for r < v/2, where r = r1 + r2 +

ft 11 yri

(j=1

+ rP. If Y1, Y2,. . ., YP are mutually

independent, then the right-hand side can be easily calculated. Branco and Dey (2001) noted that the joint pdf (5.19) is a particular case of a general class of skewed multivariate elliptical distributions. Actually, the joint pdf of the general class takes the form

fx (x)

=

ff,T (x) Fv*,T.

(AT (x

- µ))

(5.20)

where v* = v + p, T*

= T + (X - µ)T R-1 (X - Fl) , QTR-1 aaTR-1Q'

1

=

fv,T (x)

r ((v + p)/2)

f1 +

(x - µ)T R-1 (x - µ)1 v

(7rA)P/2r(v/2) (RI1/2 L

1

and

Fv' T*

(x)

(7-*), */2

=

r ((v* +1)/2)

fX

VITrr (v*/2)

(T* + y2)

dy.

00

Note that f v,, (x) is the generalized t pdf described in equation (5.6), and that Fv*,T.(x) is the cdf of a generalized version of the Student's t distribution. The mean and the variance of the univariate marginals of (5.20) are

E (X) = ar ((v - 1)/2) IT r (v/2)

7r


102

(provided v > 1) and

Var (X) =

a2 v

V

[.((v - 1)/2)12

v - 2

IF (v12)

1

(provided v > 2), respectively.

5.8 Sahu et al.'s Skewed Multivariate t Distribution Using transformation and conditioning, Sahu et al. (2000) obtained a skewed multivariate t distribution given by the joint pdf

r_V+ _P

2ta,v (x; IL, R + D2) Tin,v+,,, x

v + q(Y)

12

(I - D (R + D2) -1 D)

D (R + D2) -1

where y = x - µ, q(y) = y' (R + D2)-ly, and D is a diagonal matrix with the skewness parameters S1, ..., 5,,,,. In (5.21), tm,v(tt,1) denotes the usual m-variate t density with mean vector µ, correlation matrix 0, and degrees of freedom v. Furthermore, denotes the joint cdf of t,,,,,v(0,I). The mean and the variance of this skewed t distribution are given by

1)/ 2)

((V

E(X) = N + V

r (v/)

and

Cov(X)

=

v v

2

(R + D2)

- ii

I

F

((v(1)/2) 12 D2 F v12)

(provided v > 2), respectively, where S = (Sl, ... ,

,,.)T . The multivariate skewness measure /31,,,,, (Mardia, 1970b) can be calculated in analytic

form. The expression does not simplify and involves nonlinear interactions between the degrees of freedom (v) and the skewness parameter S when D = SI. However, /31,,,,, approaches ±1 as S -> ±oo. Sahu et al. (2000) discussed an application of this model in Bayesian regression models.

5.9 Azzalini and Capitanio's Skewed Multivariate t Distribution103

5.9 Azzalini and Capitanio's Skewed Multivariate t Distribution A slight extension of the skewed normal distribution given in (5.17) is

fy (y) _

(aTW-1(y-4)),

(5.22)

where y E RP, E E RP, W = diag( -u 1, ... , -a ), and the rest is as defined in (5.17). In the particular case = 0, (5.22) reduces to (5.17). Starting with a random vector Y having the pdf (5.22) with = 0, Azzalini and Capitanio (2002) defined a skewed t variate as the scale mixture

X=

+ Y/ /VV,

(5.23)

where vV is distributed independently of Y according to a chi-squared distribution with degrees of freedom v. Simple calculations using a preliminary result on Gamma variates show that the joint pdf of X is

(aTW_1 (x

fx(x) = 2f (x)

-)

T +_V

where Q = (x - )TR-1(x - ) and fk, Fk are as defined in (5.19). Note that this pdf coincides with that of Branco and Dey (2001) given in (5.20). In the standard case = 0 and E = R, the joint cdf of X can be represented as

Fx(x) = 2 Pr (-Uo/-,/V < 0, U/VV < x) ,

(5.24)

where (Uo, UT)T has the (p + 1)-dimensional normal distribution with zero means and covariance matrix given by 1

(S

ST

R

where

Ra 1

aaTRa

The representation (5.24) can also be written in terms of a (p + 1)dimensional t distribution. In the case = 0, simple expressions for the moments of X can be obtained. Defining Fi

F ((v - 1)/2) 8, it

F (v/2)

104


and provided that v > 1, one obtains

= Wkklik,

E (Xk)

E(X2) _ v V- 2wkk, 2

v (3 -STS) 3 E (Xk)

E Xk4)

v-3

3

wkkµk,

3v2

4

(v - 2) (v - 4) wkk,

=

E (X) = Wµ, E (XXT) =

Skewness (Xk) =

E, vv 2

v (3 - (ST(S)

v-3

Ilk

v> 2,

3v

_

3/2

V

v-2

+ 2µTµ

v-2

v>3,

T

and Kurtosis (Xk)

3v2 f (v-2)(v-4)

+6vµT v-2µ

4vµT µ (3

- 6T6)

v-3

- 3 (,5T'5)2 /

v

v-2

- µT/-6

2'

v>4. Properties concerning linear and quadratic forms of X can also be derived. For example, if a E R'' and A is a m x p constant matrix of rank m, then the affine transformation a + AX will also follow the skewed t distribution given by (5.23) with the parameters , E, and a replaced by a + E', and a', respectively (the degrees of freedom v remains unchanged), where

E'

= AEA,T

5.10 Jones' Skewed Multivariate t Distribution

W'

a

105

BTa

1 + aT (E" - B (V)_1 BT) a

W' = E', B = W-TEAT, and E" is given by E = WE"W. Also, for appropriate choices of B (a symmetric p x p matrix), the quadratic form Q = (X - ) TB (X - ) can be shown to have the f Ff,,, distribution for some degrees of freedom f. For details see Azzalini and Capitanio (1999) and and Capitanio et al. (2002). A further extension of (5.17) examined independently by Arnold and Beaver (2000) and Capitanio et al. (2002) is of the form

fy (y) = Op (y - ; E),t (ao + aT W-1 (y

- )) /41(T), (5.25)

where y E llw', T E R ao = T/ 1 - 6TR-16, and the rest are as defined in (5.22). In the particular case T = 0, (5.25) reduces to (5.22). Taking Y in (5.23) to have the pdf (5.25) with = 0, one obtains an extended skewed t distribution for X. The corresponding joint pdf for X is quite complicated, but the joint cdf can be represented as

F(x)

Pr(-(Uo+T)/V/V- 1) and

E(XXT) =

y

+ ,(T)}W8(WoT)T] IE+f772(7-)I

v2

(provided v > 2), respectively, where 77(x)

=

dxk

for k = 0,1, 2,....

5.10 Jones' Skewed Multivariate t Distribution The univariate Student's t distribution has the pdf 1+

F((v + 1)/2) v7rr(v/2)

{

x21-(-+l)/2. v

(5.26)

106


By replacing (5.26) with the skewed univariate t pdf (4.38) in a multivariate distribution, Jones (2002c) introduced a new skewed multivariate t distribution that we shall describe in this section. Let X be a p-variate random vector having the standard multivariate t distribution with the joint pdf given by

r((v + p)/2)

1 + xTx v

(v+p)/2

(5.27)

}

The univariate marginals of this are (5.26). Multiplying (5.27) by (4.38) and dividing by (5.26) yields Jones' (2002c) skewed multivariate t distribution. The corresponding joint pdf is

21-a-cr((y + p)/2)r(a + c) (v7r)(P-1)/2

a + cr((v + 1)/2)r(a)r(c) x2

(

x{1+ 1} v

a+1/2

(v+1)/2

1+

ll

X

1-

x1

a+c+xl xTx -(v+p)/2

c+1/2 (

X1

{1+`-}

a+c+x1

.

(5.28)

v J

111

This reduces to (5.27) for a = c = v/2. In the bivariate case, (5.28) is a distribution with (i) a skewed t marginal with parameters a and c in the x1 direction; (ii) conditional distributions of X2 I X1 that match those of the bivariate t distribution being t distributions on v + 1 degrees of freedom scaled by a factor of (v -x2,)/(v + 1); and (iii) a diagonal correlation matrix. Another new multivariate distribution can be obtained by replacing (5.26) by the pdf of the Gumbel distribution: exp(-x1 - exp(-x1)). This results in the joint pdf r ((v + p)/2) (v7r)(p-1)/2r ((v + 1)/2) X (1

x2

+ 1V I

exp {-x1 - exp (-x1)} xTx -(v+p)/2

(v+1)/2

{1+

l

v

}

(5.29)

With respect to the correlation structure, this pdf has much in common with (5.28). But the conditional distribution of X1 given X2, ... , Xp and the marginals are different. Jones (2001a) noted that, if Y has the beta distribution with parameters a and c, then X = a + cY/ 1 - Y2 has the skewed univariate t distribution given in (4.38). Jones (2002c) observed a similar relation-


107

(b)

(a)

Nx

-4 -2

0

2

-4 -2

4

0

2

4

X1

X1

Fig. 5.2. Jones' skewed multivariate t pdf (5.28) for p = 2 and (a) a = 6,

v=3,andc=2; and (b)a=2,v=3,andc=6 ship between the joint beta pdf

21-"-`r(a + c)F(b)

7rr(a)r(b - 1/2)I (c)

b

(1 + y1)"

(1 -

y1)1-b (1

- yi - Y2) 6-1

a>0, b>1/2, c>0 and the skewed multivariate t distribution given in (5.28) when p = 2 and b = v/2 + 1; namely, if (Y1i Y2) have the former distribution, then (X

1

7

2

)

_

(VYi a+c 1-Y12

Y2 VV +Y2(a+c-v)

1-Y2-YZ 1-Y2

has the distribution (5.28) for p = 2. In the univariate case, F and skewed t (equation (4.38)) distributions are linked in two ways that produce identical results: (i) A random

108


(a)

(b)

NX

NX

-2

0

2

-2

4

2

0

X1

4

X1

Fig. 5.3. Jones' skewed multivariate t pdf (5.29) for p = 2 and (a) v = 1; and (b) v = 20

variable with any one distribution can be obtained by transforming a random variable from the other; (ii) a random variable with each distribution can be written as a function of two independent chi-squared random variables. If Wi ti X22,,,, Fi ti F2ii,,,,,, and Ti is a random variable

with the pdf (4.38), then Ti

2

= F=

=

Ti

=

l V0

wivi

v2

Fi -

vo

vi vo

-/

(T,

Fi 2

+

wi+T,?

(5.30)


109

and

vo Wi

=

Fi

Vi WO,

where wi = vo + vi. By extending this relationship between the univari-

ate F and the skewed univariate t, Jones (2001b) introduced another skewed multivariate t distribution. It is known (see, for example, Johnson and Kotz, 1972, Chapter 40, and Hutchinson and Lai, 1990, Section

6.3) that the joint pdf of the random variables Fi, i = 1,.. . , p is vkk

r(n) r(vo)...r(VP)

f (fl, ... I M =

k=0

fj'-1

j=1

r

1v0+Ej=lvjfj)n

fi>0,...,fP>0,

(5.31)

where n = vo + + vP. Applying the transformation (5.30) to (5.31), Jones (2001b) obtained the joint pdf of Ti as f (t1, ... , tP)

=

F(n) I (v0) ... F (VP)

2 (tk +

P

wvk k

k=0

1+ P

2vk

Wk + tk)

wk + t2

(tj+ wj+ 2

2

n

Wi

j=1

ti E R,...tp E R.

(5.32)

The univariate marginals of this pdf take the form of (4.38). The conditional pdf of Ti given any subset Ti, , ... , Ti,,, of the other variables, P2 < p, is proportional to

(ti + wi + ti )

2v;

wi+ti (1+K-1 (ti+ wi+ti)

2

W1 +vi 1 +...+v'p,

where P2

K = wi

1+

2

we

(ti, +

w + ti, )

l=1

The regression of Ti given T1,.. . , Tip, takes the nonlinear form E (Ti

T'".

.

,Tip2)

=

r(

- 1/2) r (vi - 1/2) ((vi - 1/2) v 2r (o) r (vi)

vo

110

Multivariate Generalizations and Related Distributions vowi (0 - 1/2)

+ vip2 . Note that the corresponding relation where 0 = vo + vi, + for the multivariate F distribution in (5.31) is linear. If Ti,. .. ,T,,,, -

denote any m of the p Ti's (with their degrees of freedom correspondingly

renumbered as vi, ... , T1i...,TM is m F

TTTA+

along with vo), then the product moment of

wa,/2...WA /2

-

1

I-F(vi)...F(V-) Al A

ji= o

j_ = o m

xr i-1

r-\

2

i

i.1 m

vi

A,) C'.

A,

C1 A.

x

.7i-

+

2

+vo

,

provided that vi > Ai/2, i = 1, ... , m and vo > (A1 + .

+ Am)/2. In

particular, the variances and the covariances are given by

r,

1-1

wi l

,

1

2

x - v0) (vo

+

1) (Vi - 1)

F (yo - 1/2) F (vi - 1/2)

2

r (vo) F (vi)

wi-2 (vo - 1) (vi - 1)

(provided vo > 1 and vi > 1) and

Cov (Ti, Tj )

wiwjr (vi - 1/2) F (vj - 1/2) 4F (vi) F (vj)

1+vo-vi-vj

+ (vi - 1/2) (vj - 1/2) (vo - 1/2)

-(vi-u0)(ii -v0) (

F(vo-1/2)2 ) F (v0)

(provided vi > 1/2, vj > 1/2 and vo > 1), respectively. In the particular


case vo -

111

= vP = v/2, (5.32) reduces to

f (t1 ... , tP)

=

I ((p + 1)v/2)

P

rP+1 (v/2)

V + tk {(tk+v)v}

/{V+(t+)2} P

(P+1)v/2

j=1

t1 E R'. .. , tP E

(5.33)

Jones (2001b) referred to this distribution as the symmetric multivariate t distribution. Note that all of the marginals of (5.33) have the Student's t distribution with degrees of freedom v. The correlation between any two Ti's in (5.33) takes the simple form

_ 2v-3 F((v-1)/2)) 2 P

( r (v/2)

8

provided that v > 2. The limiting form of (5.32) as vo --* oo and vi > 1 remains fixed, i = 1, ... , p can be shown to be P

110ri

9 (µ1 + Q1t1i ... , µP + °PtP)

i-1 where

9 (t1, .. , tP)

P tir (2vi+1) = 2P [J (v)

exp

(- tz

8

i=1

1

2 /

F(yi-1/2) r(vi)

ILi

and

Qi

=

1

vi-1

P2

(vi -V,)1/2) r2

Note that µi and Qi are the mean and the standard deviation of 2/x2V; distribution. When v o remains fixed but. VI, .. , vP -* oo, the marginals of (5.32) tend to distribution, but the correlations between the Ti's tend to 1 and the joint distribution becomes degenerate. When all v0i v1, ... , vP -* oo, all of the marginals tend to the normal distribution

- but the form of the limiting joint distribution will depend on the


112

(b)

(a)

v N X

0 N 5

10

X1

X1

(C)

(d)

X

15

X

X1

X1

Fig. 5.4. Jones' skewed multivariate t pdf (5.32) for p = 2 and (a) vo = 2, vi = 4, and v2 = 4; (b) vo = 2, vi = 20, and v2 = 1; (c) vo = 2, vi = 1, and

v2= 20; and (d)vo=v1=v2=2

specific relationships between the v's. The limit of (5.33) as v -4 00 is the multivariate normal distribution with zero means, unit variances, and an intraclass correlation structure with correlation 1/2.

5.11 Matrix-Variate t Distribution The matrix-variate t distribution, motivated by applications in Bayesian inference, is the product of James Dickey's research in the mid-1960s. We need the following terminology to discuss its mathematical properties.

Let µ be a p x q constant matrix, let R> 0 be a p x p matrix, and let

5.11 Matrix-Variate t Distribution

113

Q>0beagxgmatrix. Form>p+q-1 define k(m,p, q) =

7r111'

rg((m - p)/2)

(5.34)

rq(m/2)

where

=

I" (X)

7rr(r-1)/4r(x)r

Cx -

2) ... r Cx

+2 2

is the generalized gamma function. Furthermore, for real or complex constants a1, ... , ap and b1, ... , bq and for random matrices S and T, define the general hypergeometric functions (see Constantine, 1963) pFq (a1.... , ap; b1, ... , bq; S)

(a,),...(ap), C. (S)

co

L.

(bl), ... (bq),,

(5.35)

k!

and

pFq (a,,...,ap;bl,...,bq;S,T) 00

FE k =o ,4

(a,),, ... (a'p), C. (S) C. (T) (bi),4 ... (bq),

(5.36)

k!

where ,={k1,...,k,,,,}, k1>k2>...>km>0,k1+k2+

W. rT(x,r.)

7r

rcr9 >

+km=k,

rm (x_ n) rm(x)

/ r(x+k1)r x+k2 - 21...r (x+k,. - r 2-

and C, (S) and C,, (T) are symmetric homogeneous polynomials of degree

k in the latent roots of S and T, respectively. A p x q random matrix X is said to have the matrix-variate t distribution with parameters µ, R, Q, and m if its joint pdf is

f (X) =

IQI(m,-p)/2 IRI-q/2

k(m,1p, q)

x Q + (X - µ)T R-1 (X - µ)

-m/2 (5.37)

(Dickey, 1966a, 1967b). If µ = 0, then we say that X has the central matrix-variate t distribution with parameters R, Q, and m. Otherwise, we refer to the distribution as a noncentral matrix-variate t. The usual multivariate t distribution (1.1) is the special case of (5.37) for p = 1

114


(single row) or q = 1 (single column). It is also known that the partic-

ular case of (5.37) for µ = 0 and R = Ip is a mixture of the normal density with zero means and covariance matrix Ip ® V - in the q x q positive definite scale matrix V. Densities of the form (5.37) appear in the frequentist approach to normal regression as the distribution of the Studentized error, both the error in the least squares estimate of the coefficients matrix and the error in the corresponding predictor of a future data array (Cornish, 1954; Kshirsagar, 1961; Kiefer and Schwartz, 1965). In the Bayesian conjugate-prior and diffuse-prior analyses for the same sampling models, it arises as the marginal prior or posterior distribution of the unknown coefficients matrix, and also as the predictive distribution of a future data array (Geisser and Cornfield, 1963; Ando and Kaufman, 1965; Geisser, 1965; Dickey, 1967b, Section 4; Zellner, 1971, Chapter 8; Press, 1972, Section 8.6). More recently, Van Dijk (1985, 1986) discussed applications of (5.37) in the linear simultaneous equation (SEM) model, which is one of the best-known models in econometrics. The SEM model is used in several areas, for instance, in microeconomic modeling for the description of the operation of a market for a particular economic commodity and in macroeconomic modeling for the description of the interrelations between a large number of macroeconomic variables.

If X has the central matrix-variate distribution with parameters R, Q and m, then it can be represented in numerous ways, as described by Dickey (1967b) and Dawid (1981). The following results (due to Dickey, 1967b, and Dickey et al., 1986) concern the conditional and the marginal distributions of X

If X = (X1, X2)T, then the conditional distribution of X1, given X2, is the matrix-variate t with parameters -R11R12 X2, Rj11, Q + X2 R22 X2, and m. If X = (XI, X2), then the conditional distribution of X1, given X2, is a matrix-variate t with parameters X2Q22 Q21, (R+X2Q22X2)-1, Q11 - Qi2Q22Q21, and m. If X = (X1, X2)T, where Xi is pi x q, then the marginal distribution of X2 is a central matrix-variate t with parameters R22 , Q and m - pl. In the particular case XT = (xi.... , xp), each row xT has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix riiQ/(m - p - q + 1). A consequence of this is that the density (5.37) of X can be written as the product of conditional


115

multivariate t distributions of the rows of X, that is,

f (X) = f (XI) f (X2 I XI) ...

f (XP I X1, ... , Xp-1)

If X = (X1, X2), where Xj is p x qj, then the marginal distribution of X2 is a central matrix-variate t with parameters R-1, Q22, and m-q1. In the particular case X = ( X I ,- .. , x9 ), each column xj has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix giiR/ (m - p - q + 1). A consequence of this is that the density (5.37) of X can be written as the product of conditional multivariate t distributions of the columns of X, that is,

f (X)

=f(XI)

(X2

X1)... f (Xq I X 1 ,

I

If X is doubly partitioned as

X=

X11

X12

X21

X22

'

where X ij is pi x qj with p1 + P2 = p and q1 + q2 = q, then the conditional distribution of X12 given X11 and X21 is a matrix-variate t with parameters R112 11 -1X11, Q11 + XT R1-1'X11, R22 - R21R111R12, and m + q1 - p- q + 1. (Here, the partitions of R and Q correspond to the partition of X.). Since this depends only on X11, it follows that X12 and X21 given X11 are conditionally independent.

The following results (due to Javier and Gupta, 1985, and Dickey et al., 1986) concern the distributions of the quadratic forms XAXT and AXB when X has the central matrix-variate t distribution with parameters R, Q, and m. If A > 0 is q x q, then the pdf of W = XAXT is given by

f (W) =

IAI-P/2

1

IRI(m-q)/2

k(m,p,q)

x IR+r

IQI-P/2 IWI(q-P-1)/2

WI--/2

x 1Fo \ 2 . (R + W)-1 W, Iq

- (QA)-1)

where W > 0, k(m, p, q) is given by (5.34), and 1Fo is as defined in (5.35). An immediate consequence of this result is that

I

W(P1)/2 IR+ WI-m/2 O

x 1Fo \ 2 . (R

+ W)-1 W, 1q- (QA)-1) dW

116


= F (q/2)Ir

q)/2) IAIp/2 IRI-(m-q)/2

jQjp/2

Hence the hth moment of I W I is

_ F ((q + 2h)/2)1' ((m - q - 2h)/2)

[IWIh]

E

1' (m/2) x IAIp/2 IRI-(m-q-2h)/2 IQIp/2.

(5.38)

Further using the fact that an F-distribution is uniquely determined by its moments, it follows that I W I can be written as a product of q independent univariate F's, that is, q

IWI

^'

rIF'(q-(j-q),m-q-(j-1)) j=1

For the special case A = Ip and p = q, (5.38) gives the hth moment of XXT.

If A > 0 is p x p and B > 0 is q x q, then AXB has the central matrix-variate t distribution with parameters BTQB, AR-'A T , and

m,m>p+q-1.

If A > 0 is p x p and B is a q x r rectangular matrix, then AXB has the central matrix-variate t distribution with parameters BTQB, AR-1A T , and m, m > p + r - 1. If a is a q x 1 vector, then aT XT has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix a TQaR/(m - p - q + 1). If a is a q x 1 vector such that aTa = 1 and b is a p x 1 vector, then aT XT b is a linear combination of Student's t random variables. If b is a p x 1 vector, then XTb has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix

bTRbQ/(m-p-q+ 1). If a is a q x 1 vector and b is a p x 1 vector, then

(m - p - q + 1)a TXT b (aTQa) (bTRb) has the Student's t distribution with degrees of freedom m - p - q + 1.

In the special case R = Ip and Q = 1, if a is a real number and b is a q x 1 vector such that a2bTb = 1, then aXb has the Student's t distribution with degrees of freedom m - q. Javier and Gupta (1985) also derived a useful factorization of the central matrix-variate t density in terms of the product of q - 1 independent


117

univariate F densities and q independent multivariate t densities - paralleling the result of Tan (1969a) for matrix-variate beta distributions. Let X be a p x q random matrix having the density (5.37) with It = 0. Set

U=

(R-1/2XQ-1/2) (R-1/2XQ-1/2)T

so that U is p x p, symmetric, and U > 0. Partition U as

U =

CU11

U21

U12 U22

so that U11 is 1 x 1 and U22 is (p - 1) x (p - 1). Abbreviating D22 D21D111D12 by D22.1, define the following submatrices U2221

=

U22.11) 22.1

U22.1

-

U 11

=

( U22.11) 11

U22.1, j = 1 2,...,p - 1,

and

Ui°)

= U11,

so that is (p - j) x (p - j) and U11) is 1 x 1. With all of this notation the factorization of the density of X (due to Javier and Gupta) can be stated as q-2 m-jq-2(p-.7)-2 f(X) _ 11 F p-j+2 2 2 )

j=0

P

X

1I tUii) {1+uM

(j

IP-j + U22,1; m - (j - 1) q

j=1

where t (T; r) is the joint pdf of a central multivariate t distribution with degrees of freedom v and correlation matrix T and F(a, /j) is the pdf of a univariate F distribution with degrees of freedom a and /3. The two predictivistic characterizations of the multivariate t distribution based on (5.14) and (5.15) have the following matrix-variate generalizations Let X1, X2, ... be an infinite sequence of q-dimensional random column vectors that are orthogonally invariant (which means that, for

118


each k, X(k) = (X1i ... , Xk)T and rX(k) are identically distributed,

for all k x k orthogonal matrices I') and, for k fixed, let X ik)T , i = 1, 2,.... If XI, ... , Xq are linearly independent with probability 1 and

E [X(q)TX2(q) I Xiq)1

= aX(q) TX(q) + B

where 0 < a < 1 and B is a q x q positive definite matrix, then the distribution of X(P) is the matrix-variate t with µ = 0, R = IP,

Q=(1/a)B,andm=1+(p/a)-p. Let X1, X2, ... be an infinite sequence of q-dimensional random col-

umn vectors such that, for each p, X(P) and rX(P) are identically distributed, for all p x p orthogonal matrices r satisfying rip = 1P (where 1P is a p-dimensional vector of l's). Under this assumption there exists a u-algebra T of events such that n

1 Xi Xn = -1: n i=1

E(Xi IT)=M and

n

Sn

=

n i-1

(Xi - Xn) (Xi - Xn)T

E (XIXT I T) - E (XI I T) {E (XT I T) }T = V as n -* oo (Chow and Teicher, 1978), where the convergence is almost everywhere. Moreover, if

E [(xv - 1gMT)T (X(q) - 1g MT) I X(q) M] a (X(q)

- 1gMT)T (X(q) - 1gMT) + B, q\\

where 0 < a < and B is a q x symmetric positive definite matrix, then X(P) is a location mixture of the matrix-variate t distribution 1\1

with µ = 1TM, R = Ii,, Q = (1/a)B, and in = 1 + (p/a) - p. In addition, M and V are independent. Dawid (1981) provided a different but more convenient parameterization of (5.37). If Y (p x p) has the standard matrix inverse Wishart distribution with parameter 8 and if, given Y, X (n x p) has the matrix normal distribution with parameters In and Y, then X is termed

5.12 Complex Multivariate t Distribution

119

as having the standard matrix t distribution. In the notation of (5.37), this would correspond top = 0, R = I, , Q = In, and m = 6 + n + p - 1. Under Dawid's parameterization, if X* is a n* x p* submatrix of X, then X * has the matrix t distribution with parameters In-, Ir. , and S: Note that 6 is unchanged. This kind of consistency enabled Dawid (1981) to construct what is termed as the standard infinite matrix t distribution. Namely, X = {x2j, i > 1, j > 1} is said to have the above-named distribution if it has the property that for all (n, p) the leading n x p submatrix of X has the standard matrix t distribution with parameter 6. The standard matrix t distribution also has the attractive property of being spherical, that is, if P (n x n) and Q (p x p) are two orthogonal matrices, then both PX and XQ have the same distribution as X.

5.12 Complex Multivariate t Distribution A complex normal random vector Y = V +W is a complex random variable whose real and imaginary parts possess the bivariate normal distribution. A complex p-variate normal random vector

Y = v+Tw (Vl+

Wl, V2+ JW2i...,VP+

WT (5.39)

is a p-tuple of complex normal random vectors such that the vector of real and imaginary parts (Vi, Wl,... , Vr, Wr) has the 2p-variate normal distribution (Goodman, 1963). Section 45.13 of Kotz et al. (2000) provides an account of this distribution. It is usually assumed that the 2p-variate normal distribution of (V1, Wl,... , VP, Wr) has zero means and covariance matrix given by

1 (E1 -E2 1

2 ` E2

E1

where E1 is symmetric (matrix A is symmetric if AT = A) and E2 is skew-symmetric (matrix A is skew-symmetric if A = -AT) . From the given structure it is easily seen that the covariances of the p-variate vectors V and W are each equal to E1/2 and the covariance between V and W is equal to E2/2. Hence the covariance of the complex p-variate normal random vector Y in (5.39) is E1 + V/-IE2 = E, say. The properties of the distribution of Y have been studied by many authors. The joint pdf of Y is given by

f (Y) =

7rp IEI

eXP {-yTE-ly} ,

(5.40)

120


where y denotes the complex conjugate of y (Goodman, 1963). For ex-

ample, for the complex univariate normal distribution, y = vl +wi and the covariance matrix E = a2, and thus the joint pdf of Y becomes

vi +wi

1

a2

The characteristic function of Y can be shown to be

E [exp {i (sTV + tTW) }] = exp { - uTEu}

where u = s +t (Wooding, 1956). Explicit expressions for the moments of Y have been derived by Sultan and Tracy (1996). The complex multivariate normal distribution has applications in describing the statistical variability of estimators for the spectral density matrix of a multiple stationary normal time series and in describing the statistical variability of estimators for functions of the elements of a spectral density matrix of a multiple stationary normal time series. Relatively few results are available that deal with complex multivariate t distributions. Originally, the complex multivariate t distribution was introduced by Gupta (1964). Let Y have the complex p-variate normal distribution with zero means, common variance Q2, and covariance matrix o, 2R. Let 2vS2/Q2 have the chi-squared distribution with degrees of freedom 2v, distributed independently of Y. Then X = Y/S is said to have the complex p-variate t distribution with degrees of freedom v and correlation matrix R. By writing down the joint distribution of S and X and then integrating out S, the pdf of X can be obtained as ( v+P)

f (x)

_

IIRIP

{1+

1XTR-lx }

Tan (1973) discussed some properties of this distribution. Tan (1969b) provided a brief discussion of a complex analog of the matrix-variate t distribution given by (5.37).

5.13 Steyn's Nonnormal Distributions Strictly speaking, this section does not deal with multivariate t distributions per se. This section is about nonnormal distributions arising from the class of multivariate elliptical distributions that contains the multivariate t as a particular case. One weakness of the class of multivariate elliptical distributions is that

5.13 Steyn's Nonnormal Distributions

121

all fourth-order cumulants are expressed in terms of a single kurtosis parameter (moreover, the univariate marginals have zero skewness and the same kurtosis). In fact, the cumulant generating function (cgf) and the moment generating function (mgf) of a p-variate elliptical distribution with zero means and correlation matrix..R are K (t1,...,tp) 1

(tTRt) + 2 r { 2 (tTRt) } + >2 Ak (tT Rt) k l

(5.41)

k> 3

and M (tl,...,tp) /

(tTRt) 2

tTRt l

exp

2

I

/

\\\

1+2

1!

Bk (tT Rt) k

+

(5.42)

,

k>3

respectively, where Ak, Bk are constants and n is the kurtosis parameter. Steyn (1993) attempted to introduce meaningful multivariate distribu-

tions that are related to the elliptical distributions and that contain more than one kurtosis parameter. As an example, consider a random vector (X1, X2, X3) possessing the three-dimensional normal distribution with the mgf M (t1, t2, t3)

(t1 + t2 + t3 + 2r12t1t2 + 2r13t1t3 + 2r23t2t3)

exp

.

1

2

(5.43)

Suppose this model is placed in a changing environment that favors a change in one of the random variables, say X1, in such a way that the kurtosis should be taken into consideration. Specifically, assume that the marginal distribution of X1 is elliptical with the kurtosis parameter i1, while the conditional distribution of (X2, X3) given Xl = x1 remains unchanged. Note that (5.43) can be written as M (tl, t2, tr3) 1 2=

exp L1

J

ll L(1-ri2)t2+(1-ri3)t3+2(r23 -r12r13)t1t2}1

LLLL 2

/ XJ

2 l r exp S - 2 + (ti + r12t2 + r13t3) xl y dxl. ll

(5.44)

JJJ

Changing the probability element in the integrand in (5.44) to that of

122


the elliptical distribution in (5.42), one can show that the mgf changes to

= M (tl, t2, t3) exp 1 g (t1 + r12t2 + r13t3)4 + ...1

M1 (t1, t2, t3)

.

The corresponding cgf becomes K1 (t1, t2, t3)

2 (1 + t2 + t3 + 2r12t1t2 + 2r13t1t3 + 2r23t2t3)

+ 1 K1 (tl + r12t2 + r13t3)4 +

(5.45)

Setting t2 = t3 = 0, the cgf of the marginal distribution of X1 is given by

K1 (t1,

0,0)

=

2t1 +

.. .

gr.lt1

which shows (as it should) that the marginal distribution of Xl is elliptical with kurtosis parameter is (compare with equation (5.41)). However, for tl = t3 = 0 and tl = t2 = 0, one obtains Kl (0, t2, 0)

K1 (0, 0, t3)

=

=

2t22 + 18K1

(r12t2)4 + .. .

2t3 + 8n1 (r13t3)4 +

thus, the marginal distributions of X2 and X3 are also elliptical but with kurtosis parameters i1r12 and rjr13, respectively. Furthermore, for ti = 0, K1 (0,t2,t3) = 2 (t2 + 2r23t2t3 + t3) + 81

r13t3)4 + ...

(r12t2 1

which shows that the joint marginal distribution of (X2, X3) is not elliptical. The fourth-order cumulants are easily obtained from (5.45) as K2Jk = 3K1r12r13, where i + j + k = 4. Suppose now that the model given by (5.43) is placed in an environ-

ment that favors a change in not only Xl but also influences (X2, X3). Assume - in particular - that the conditional distributions of X2 given Xl = xl and X3 given (X1,X2) = (XI, X2) are elliptical with kurtosis parameters K2 and ic3, respectively. Then calculations similar to those above show that the mgf (5.43) changes to 2

M2 (tl, t2, t3)

= M (tl, tz, t3) exp

2

2

z 2

2

2

2

1

I


/2 +2r13

\2

2

t3 2.12 )

I

123

+...

(5.46)

where u1

u2

=

tl + r12t2 + r13t3,

= t2 + r23 - r12r13 t3, 1 - rig 2

Q2.1

=

2

1 - T12,

and 2

0'3.12

1-r132

(r23 - r12r13)2

1-r212

It is easily seen that the marginal distributions of X1, X2, and X3 are elliptical with kurtosis parameters given by n1, r12r.1 + o"2.1K2, and rl3r'1 + (Q3.1 - a3.12)t3, where Q3.1 = 1-r13. This time, the fourth-order cumulants are given by ICzj/

= 3 t IGlri2r13 + K20`2.,

r12r13)kI

(r23 -

,

where i + j + k = 4. In the case of '004, should be added. XP)T has Similar constructions can be performed when X = (X1' a p-variate normal distribution with zero means, covariance matrix R, and the corresponding mgf

= exp 12 tTRt)

M2 (t1, ... , tp)

.

(5.47)

Consider two environments similar to those considered above for the trivariate normal model. First, divide X into two random vectors X(1) _ (X1, ... , Xh)T and X(2) = (Xh+1, ... , XP)T , and let

R = C

R11

R12 )

R12

R22

be the corresponding partition of the correlation matrix. Also let t(l) = ( t 1 ,- .. , th)T and t(2) = (th+1 i ... , tP)T be the corresponding partition

of t. Now assume that the marginal distribution of X(1) is changed to an h-dimensional elliptical distribution with kurtosis parameter k


124

(a)

(b)

Nx

Nx

-3

-1

X1

X1

(C)

(d)

V

1

3

1

3

V

-3 X1

-1

X1

Fig. 5.5. Steyn's bivariate pdf corresponding to (5.46) for t3 = 0 and (a) rcl = 0.8, K2 = -0.4, and r12 = 0.2; (b) i1 = 0.8, ice = -0.4, and r12 = 0.8; (c) Kl = -0.4, K2 = 0.8, and r12 = 0.2; and (d) icl = -0.4, K2 = 0.8, and r12 = 0.8

and that the conditional distribution of X(z) given X(1) = x(1) remains unchanged. Then calculations show that (5.47) changes to M1(t)

= M(t) exp

T

(t(1) 8

+ R111R12t(z)/ z

xR11 (t(1) + R111R12t(2))

+.

.

(5.48)

Clearly, z

M1 (t(1), 0)

= exp {t(1)TRllt(1) + 8'c1 (t(1)TR11t(1)) +...


125

and

M1

(p

t(2))

= exp

2it(2)TR22t(2) 2

+1K, (t(2)TR22R111R12t(2) )

+...

which shows that the marginal distribution of X(1) is an h-dimensional elliptical distribution (as it should) while that of X(1) is not elliptical. The second-order cumulants of (5.48) are the same as those for (5.47). For the second construction, partition X2 into X(3) = (Xh,... Xh+s)T and X(4) = (Xh+s+1, ... , Xr)T, and let t(2) be partitioned correspondingly into t(3) and t(4). Let C denote the conditional covariance matrix of X2 given X1 = x1, that is, )

C = R22 - R.21R111R.12,

and let C be partitioned as

C= C

C11

C12

C12

C22

so that C11 is s x s, C12 is s x (p-h-s), and C22 is (p-h-s) x (p-h-s). Now, assuming that the distribution of X1 is elliptical with kurtosis parameter i1 and that of X(3) - E(X(3) I X(1) = x(1)) is elliptical with kurtosis parameter KK2, one can show that the mgf (5.47) changes to

M2(t)

= M(t) exp [({ (t(1) + R111R12t(2) )

T

2

xR11 (t(1) + R111R12t(2)) I +

+ 8k2 (t(3) +

T

C_11C12t(4)

) 2

xC11 (t(3) + C11C12t(4))

+ ... (5.49)

This defines the mgf of a multivariate distribution that is equal to the product of the mgf of the multivariate normal and a function of two quadratic forms in t depending on the two kurtosis parameters iGi, i =

126


1, 2, and on the elements of the normal covariance matrix. Setting t(2) = 0 into (5.49), we see that X(1) has an h-dimensional elliptical distribution with zero means, covariance matrix R11, and kurtosis parameter Ic1 (as

it should). If either t(1) = 0 or t(1) = 0 and t(3) = 0, then M2(t) becomes a function of three different forms.

5.14 Inverted Dirichlet Distribution There is a close connection between the multivariate t distribution defined by (1.1) and the inverted Dirichlet distribution (Cornish, 1954; Dunnett and Sobel, 1954). To see this, consider the central p-variate t distribution with the pdf XTR'-1xl -(v+p)/2

r ((y + p)/2)

=

f (x)

(7fv)p/2r (v/2) IRI

1/2

1+

JII

V

Upon transforming to the canonical variables Z = (Z1, ..., Zp), Z = PX, where P is a p x p matrix such that pTp = R-1, it is easily seen that f (z)

=

F ((v + p)/2) 1 + zTZI (v+p)/2 (7rv)p/2r (v/2) L VJ

(5.50)

In (5.50) now perform a further transformation Ti = ZZ iv, which is one-to-one in each of 2p regions with the Jacobian IJI

= 2-pvpl2/

p.

Consequently, the joint pdf of TT = (T1, . . . , Tp) becomes P

f (t) =

r ((v + p)/2) t-1/2 ... t-1/2 (1 + p 1 7fp/2F (/2) i=1

(v+p)l2

E ti)

which is the inverted p-dimensional Dirichlet distribution D'(1/2, ..., 1/2; v/2); see, for example, Kotz et al. (2000, Chapter 49).

6

Probability Integrals

There has been a very substantial amount of research carried out on probability integrals of multivariate t distributions. Most of the work was done during the pre-computer era, but recently several computer programs have been written to evaluate probability integrals. Sections 6.1 to 6.7 by now may have lost some of their usefulness but are still of substantial historical interest in addition to their mathematical value. We have decided to record these results in some detail in this book in spite of the fact that some of the expressions are quite lengthy and cumbersome. Sections 6.8 to 6.13 contain more practically relevant and modern results.

6.1 Dunnett and Sobel's Probability Integrals one of the earliest results on probability integrals is that due to Dunnett and Sobel (1954). Let (X1, X2) have the central bivariate t distribution with degrees of freedom v and the equicorrelation structure r2j = p, i $ j. The corresponding bivariate pdf is

f(xi,x2) v,P) =

{'+ xl + x2 - 2pxlx2 l

1

2 1 - p2

v(1 - p2)

}

with the probability integral P (yi, y2; v, P)

=

f (xi, x2; v, P)dxldx2.

Let

x(m,yi,y2)

(Yi - PY2)2

_

)2 + (1 - PZ)(m + y(yl - Pyz

127

(v+2)/2 (6.1)


128

and let

f

(a, b)

x( m,yi,y2)

r(a

+ b) a-1 rr(b) y (1- y)

1

dy

denote the incomplete beta function. Dunnett and Sobel (1954) evaluated exact expressions for (6.2) when v takes on positive integer values. For even v and odd v, they obtained P(y1, y2; v, P)

=

1 arctan

1-P

2

-P

27r

F(j - 1/2)

+ y2 vir jj=1

4

1/2-j

+ y2 v

Cl

1

rr

x L1 + sgn(yl - PY2)Ix(v,y,,y2)

vf2 r(j - 1/2) (1 + y1

+ y1 4

v7r

x

1:

r(j)

j=1

11

1

2,7 -

2

12-j

v (21,

+ sgn(y2 - PY1)Ix(v,y2,y,)

j-2)J

and

P(y1, y2; v, P)

=

27r

±vaai }

arctan { -VvL

+

(v-1)/2

y2

4

v-7r

E j=1

+

r(j)

r(j + 1/2) (

-7

y2

v 1

x [1 + sgn(yl - PY2)Ix(v,yl,y2) (v-1)/2

+ y1 4

vir

E j=1

I(j)/ r(j+1

r

j)

1+y1-j

2)

v/

x I1+sgn(y2-Py1)Ix(v,y2,y,) L

2

G 'j) (6.4)

respectively. Here, a = y1 + y2, # = Y1Y2 + PV,

6.1 Dunnett and Sobel's Probability Integrals

129

7 = y1y2 - v,

and 6

= Vyi - 2Pyiy2 + y2 + v(1 - p2).

In the special case yl = Y2 = 0, both (6.3) and (6.4) reduce to the neat expression

= arctan

P(0, 0; v, p)

__p

2

V1

(6.5)

,

_p

which is independent of v and is therefore identical with the corresponding result for the bivariate normal integral. Since the number of terms in (6.3) and (6.4) increases with v, the usefulness of these expressions is confined to small values of v. Dunnett and Sobel (1954) also derived an asymptotic expansion in powers of 1/v, the first few terms of which

yield a good approximation to the probability integral even for moderately small values of v. The method of derivation is essentially the same as that used by Fisher (1925) to approximate the probability integral of the univariate Student's t distribution: Express the difference f (xl, x2; v, p) - f (xl, x2; oo, p) as a power series in 1/v and then integrate this series term by term over the desired region of integration. Setting r

2

=

yi - 2Pyly2 + y2

1-P2

1

Dunnett and Sobel obtained .f (y1, y2; v, P) .f (Y1, y2; 00, P)

r2+ 4

-r) 2

r12

+

(r8

1

5r6

32

1/

7r10

12

13r8

4

8

96 + 24 - r)

( 384 r16

( 6144

r14

17r12

128

144

1

+r) v2 1

v3

77r10 120

1

r) v4 8

+

1 + D(r),

f

say. Thus, the desired probability integral is P(y1, y2; v, P)

=

f (xl, x2; oo, P)dxl dx2 00

CO

/92

/91

J D(r)f(xi,x2;oo,P)dxldx2. (6.6) -00 -00

+J

The first term on the right-hand side of (6.6) is the integral of the bivariate normal pdf, and it has been tabulated by Pearson (1931) with


130

a series of correction terms. The second term can be integrated term by term to obtain an asymptotic expansion in powers of 1/v. Dunnett and Sobel gave expressions for the coefficients Ak of the terms 1/v' for k = 1, 2, 3, 4. The first of these coefficients takes the form

Al =

a42O(a)0(y2)

+

b410(b)O

(yi) -

(y2) ID (a)

_yi(y1 4

where 0 and 4) are, respectively, the pdf and the cdf of the standard normal distribution, and

a =

Y1 - Py2

1-p2' y2 - Py1

1-p2 In the special case y1 = y2 = y, (6.6) reduces to V

P(y, y; v, P)

V

= f f f (xi,00x2; oo, P)dxldx2 + `41 +

A2

V

v2

+

A3

+

v3

A4

+

.. .

(6.7)

V4

with the first two coefficients Al and A2 now taking the forms Al

=

YO(y) J (y2

+ 1) (cy) -

y(p'(cy)}

and A2

y4(y) { (3ys

- 7y4 -5 Y2 - 3) (D (cy)

-y-'(cy) [3y4 (c4 + 3c2 + 3) - y2 (c2 + 5) - 3] },

where c =

(1 - p)/(1 + p). In this special case, Dunnett and Sobel

(1954) tabulated numerical values of the coefficients Ak for selected values of p, y, and v. The following table gives the values for p = 0.5

6.2 Gupta and Sobel's Probability Integrals

131

Coefficients of the asymptotic expansion (6.7) for p = 0.5 y

v

Al

A2

A3

A4

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

4

0.003371 0.008999 0.021983 0.047374 0.079687 0.096254 0.067469 -0.020268 -0.149011 -0.276255

3.00

18

-0.025870 -0.057784 -0.100016 -0.150182 -0.198378 -0.231628 -0.240531 -0.223682 -0.187525 -0.142571 -0.062685

0.003816 0.006868 0.006891 -0.006835 -0.033130 -0.038696 0.052274 0.293449 0.623867 0.858993 0.432592

-0.001050 -0.002155 -0.001879 0.007991 0.036817 0.032808 -0.191482 -0.819219 -1.618705 -1.765249 2.236773

4 6 5

6 6 9

12 13 12

-0.376815

These values can be used to construct tables for the probability integral in (6.7).

6.2 Gupta and Sobel's Probability Integrals Gupta and Sobel (1957) investigated the special case when X follows the central p-variate t distribution with degrees of freedom v and the correlation structure r2j = p = 1/2, i 54 j. If Yl, Y2 i ... , Y,,, Y are independent normal random variables with common mean and common variance v2, and if vS2/o2 is a chi-squared random variable with degrees of freedom v, independent of Y1, Y2,. . ., Y, , Y, then one can rewrite the probability integral as

P(d)

=J

d

/'d

.. . J

f(xl,...,xp;v,p)dxp...dxl

00

= Pr max(Y1iY2, S...,Yp) -Y

< vf2-d1

Pr(Mv; Y 0, 7r + arctan (y2/y1) , if yl < 0, A1

=

A2

=

1

1+p'

and 1

1-p

One of the series formulas given is

1rO

P-

(-c)k

r((v + k)/2)

2 fF(v/2) O (1 + yl + y2)"/2 r((1 + k)/2) 02

X

cosk(9 - O)d9.

(6.16)

fol

For the special case v = 1, P can be reduced to the closed-form expression

where 2r sin 0

U

A(1+r2+2rcos =

V

r

1-r2 A(1 +r2 +2rcos0)' Yi + 'Y2

=

1+

1+yl +y2

,

and

A =

P =

1 21r

arctan

tan

(02_lr)

Y1Y2

(V'1+yi +y2

2

7r

+ arctan y1 + arctan y2 +

2

The advantage of these expressions over the ones given by Dunnett and

Sobel (1954) is that these are easier to compute, especially for large

6.5 Steffens' Noncentral Probabilities

139

degrees of freedom. For instance, the integral in 0 in (6.16) can be expressed in terms of incomplete beta functions that are extensively tabulated. Amos and Bulgren (1969) numerically evaluated values of P for all combinations of p = -0.9, -0.5,0,0.5,0.9 and v = 1, 2,5, 10, 25, 50.

6.5 Steffens' Noncentral Probabilities Consider the p-variate noncentral t distribution defined in (5.1). Motivated by the Studentized maximum and minimum modulus tests, Stef-

fens (1970) studied the particular case for p = 2 and R = Ip. In this case, the joint pdf (5.1) reduces to f (X1, X2)

= exp

( ei +e21

X

7rF (v/2)

2

f

°°

1

Cl

k

V LSlxl 1

k!l!V(k+1)/2+1

k=0 l=0 l

(V LS2x2)

°° r((v+k+l)/2+1)

(1 +

2v+ x2v (v+k+l+2)/2 J

where ei = µj/o, are the noncentrality parameters and v denotes the degrees of freedom. The testing procedures involve maximum or minimum values of the components X1 and X2 and the computation of the corresponding probabilities. For this reason, Steffens (1970) derived series representations for probabilities of the form P1 = Pr(I X1 1< A, I X2 1< A) and P2 = Pr(I X1 J> A, I X2 > A). It is seen that P1

= 2exp -

2+2 2

00

)

00

1: E

(e2/2)k (e2/2)l

k!l!B (k + 1/2,1 + 1/2) k=o t=o

7r/4

( sin2k

x fo

v cos21 v+ sin21 v cos2k v)

xIa (k+1+1, 2)dv and P2

=

2(exp

i

+L) 2

00

CO

(f2/2)l EE k!l!B (k + 1/2,1 + 1/2) k=O1=0

fr/4 x

(sin2k v cos21 v+ sin2t v cos2k v) x

{1-Ip (k+1+1, 2)}dv,

where I, denotes the incomplete beta function ratio, a = A2 sec2 v/(v+ A2 sec2 v), and /3 = A2cosec2v/(v + A2cosec2v). Using these represen-


140

tations, Steffens estimated values of the critical points A for all combinations of v = 1,2, 5,10, 20, 50, oo and 1i T=2 = 0(1)5 for the significance level 0.05. In a more recent development, Bohrer et al. (1982) developed a flexible algorithm to compute probabilities of the form Pr(cii _ < XP < c21, ... , clp _< Xp < c2p) associated with the noncentral p-variate distribution (5.1).

6.6 Dutt's Probability Integrals Dutt (1975) obtained a Fourier transform representation for the probability integral of a central p-variate t distribution with degrees of freedom v and correlation matrix R

P

(y1,...,yp)

yl f...ff(xi,...,xv;v)dxp...dxi.(6.17)

_

' OPO

Using the definition of multivariate t, one can rewrite (6.17) as 00

2

P (yl,...,yp)

J

zv-1 exp (-z2/2) G (yl, ... , yp) dz, (6.18)

where yk = ykz/-\,fv-, k = 1,... , p and G is the joint cdf of the multivari-

ate normal distribution with zero means and correlation matrix R. In the case yk = 0, one has P independent of v and

P(y1,...,yp)

= G(0,...,0).

Explicit forms of G for p = 2, 3, 4 in terms of the D-functions are given in

Dutt (1973). The D-functions are integral forms over (-oo, oo) defined by 00

Dk (t1, ... , tp; R)

(27r)k

dk

f.... foo Si...Sk

(itlsl _s k

xexp

l-0

2

dsk

dsl,

l=0

where the first five dk are d1

=

1,

d2

=

d12,

d3

= =

d12+13+23 - (d12 + d13 + d23)

d4

-d12+13+23+14+24+34 + d12+13+23 + d12+14+24 + d13+14+34

6.6 Dutt's Probability Integrals

141

+d23+24+34 - (d12 + d13 + d23 + d14 + d24 + d34) d5

=

-dl2+13+23+24+34+15+25+35+45 + d12+13+23+14+24+34 +d12+13+23+15+25+35 + d12+14+24+15+25+45 +d13+14+34+15+35+45 + d23+24+34+25+35+45

- (d12+13+23 + d12+14+24 + d12+15+25 + d13+14+34 +d13+15+35 + d14+15+45 + d23+24+34 + d23+25+35

+d24+25+45 + d34+35+45) + d12 + d13 + ... + d45,

and dpi 9i +...+pm 9m

=

1 - exp {- (rpi 9i spi S9,

+ ... + rpm qm Spm S9m) }

Using the notation Dk: jl ,...,jk

Dk {tj1,...,tjk; R (tjl,...,tjk)},

where R (tj1, ... , tjk) is the correlation matrix based on the subscripts jl, ... , jk, Dutt (1973) provided the following explicit forms for G G (tl, t2) G (t1i t2, t3)

=

=

{1 - 4' (tl)} {1 -'t (t2)} + D2:1,2,

{1 - (D (t1)} {1 -"D (t2)} {1 -'D (t3)} + {1 - 1 (t1)} D2:2,3 + {1 - 1 (t2)} D2:1,3 + {1- 4 (t3)} D2:1,2 + D3:1,2,3,

and 4

G (t1, t2, t3, t4)

_

11 {1 -

(tk)} + 11 - 4' (tl)} {1 -'P (t2)1 D2:3,4

k=1 +{1-'I(tl)}{1-4 (t3)1 D2:2,4

+{1-4) (t2)}{1-'D (t3)} D2:1,4

+{1-c1(tl)}{1-$(t4)}D2:2,3 + {1 - 4) (t2)} 11-4) (t4)} D2:1,3 + {1 - 4) (t3)} {1 - (D (t4)} D2:1,2 + {1 - 4) (t1)} D3:2,3,4 {1 - 4) (t2)} D3:1,3,4

+ {1 - 4) (t3)} D3:1,2,4 + {1 - 4) (t4)} D3:1,2,3 +D4:1,2,3,4

A much simplified representation for G in terms of the error function, erf(.), and integral forms over (0, oo), denoted as the D* functions, is


142

given in a later paper by Dutt (1975). These D*-functions are defined by

Dk tl,...,tP;R

[00

2

(27r)k

... [00 -d*k k

(_s?/2) dsk " ' dsl

x exp

,

(6.19)

1=0

where for the first few k are di

= sin(tisi),

d2

= e_12 cos1_2 -e12 cos1+2,

d3

= el2+13+23+14+24+34 COS1+2+3+4 +e12-13-23-14-24+34 COS_1_2+3+4

+e-12+13-23-14+24-34 COS_1+2-3+4 +e-12-13+23+14-24-34 COSl_2_3+4 -e-12-13+23-14+24+34COS_1+2+3+4 -e-12+13-23+14-24+34 COS1_2+3+4 -e12-13-23+14+24-34 COS1+2+3+4

-el2+13+23-14-24-34 COS1+2+3-4

and for notation ePl gl+...+p-q-

exp { - (rp1 q1 sp1 sq1 + ... + rpm q,n spm sq,,,) J ,

= COSP1+ +P-

sin (tP1 sP1 + ... + tpm Sp_) ,

= COS (tn, sp, + ... + tPm sp-)

(A negative sign on the index pl ql corresponds to +rp1 q, sp1 sq1 and -pl corresponds to -ti, sp, .) Important special cases of these functions are Di (y)

=

1 erf 2

D2 (0, 0; R) =

y V'(2

1 arcsin (r12) ,

and D* (0; R)

0,

=-

for k odd.

Using the abbreviation that Dk:jl,...,jk

-

Dk 1.

21'

S ..

t3k ; R (tjl, ... , tjk )}

,

6.7 Amos' Probability Integral

143

Pp-22 P

Dutt (1975) provided the following representation for G

(1) mp

G(t1,...,tP)

Di:k+

k=1

(j)p-3

\

D2:kl

k c 11 z 11}, 11 z 11= v7 z, and c is a nonnegative

constant. The set E(c) is the cone, with vertex at the origin, which intersects origin-centered spheres in spherical caps. This is illustrated in Figure 6.1 for p = 2. Bohrer (1973) studied the analytical shape of A, (c) and the associated probability p(c, r, p, v)

= Pr (X E A,.(c))

when X has the p-variate t distribution with mean vector 0, covariance matrix o,2Ip, and degrees of freedom v. The evaluation of p(c, r, p, v) is of statistical interest and use in the construction of confidence bounds (Wynn and Bloomfield, 1971, Section 3; Bohrer and Francis, 1972, equation (2.3)) and in testing multivariate hypotheses (Kudo, 1963, Theorem 3.1, Section 5; Barlow et al., 1972, pages 136ff, 177).

6.9 Probabilities of Cone

147

As regards the shape, Bohrer showed that every two-dimensional section of Ar containing the zl-axis is exactly the two-dimensional version of Ar illustrated in Figure 6.1. Thus, Ar is the solid of revolution about

the zl-axis that is swept out by the Ar in Figure 6.1. To express this more precisely in mathematical terms - for an p x 1 vector v - define polar coordinates R and µv = {Bvi}, with -7r < Ovi < 7r, by

= Rvcos0 i,

v1

i-1

Rv cos Bvi [J sin 9Vj,

Vi

j=1 i = 2,...,p - 1,

and i-1

VP

= RV 1 sin9vj. j-1

Also define = arccosc,

0*

T1

= {x : Bt1 - 0* E (0, 7r/2], Rt cos (0t1 - 9*)
0, j = 1, 2, ... , p under a p-variate t distribution with v degrees of freedom can be expressed in terms of the function V, (Y1, Y2,

yp), the

integral of the p-variate t pdf f (x1, X2.... ) xp; v) 1+x1+x2+...+xP

F((v+p)/2) (v7r)pl2F(v/2)

-(v+p)l2

v

over the same p-dimensional convex polyhedra. John also provided an important asymptotic expansion in powers of 1/v connecting Vv(yl, y2i . .., yp) with V (y1i y2i ... , yp), the integral of the p-variate normal pdf f (xi, X2.... , xp; oo)

_

(27r)-p/2

exp {- (xi + x2 + ... + xP) /2}

6.10 Probabilities of Convex Polyhedra

151

over the same polyhedra discussed above. Up to the order of the term O(1/v2), the expansion is V (y1, y2, ...

,

Yp)

V (y1, y2, ... , Yp) +

4v

1y1y2f (y1, y2) V (y3, y4, ... , Yp)

-yl (1 + yi) f (y1) V (y2, y3, ... , yp) } +96v2 {3y1y2y3y4f (y1, y2, y3, y4) V W5, ys, ... , yp) -Y1Y2Y3 (2 + 9yi +6 Y2 + 3y3) f (y1, y2, y3) V (y4, ... , yp)

-Y1Y2 (3 + 5yi + y2 - 9yi - 9y1 2 - 3y2) XV (y3,...,yp) f (Y1, Y2)

+yl (3 + 5yi +7 Y4

- 3y6) f (YO V (y2, ...

Yp)

}

+olv/. In this formula, V (y,,, y,,.,,+1, ... , yp) is to be replaced by 0 if m > p + 2

and by 1 if m = p + 1. In principle, there is no difficulty in determining further terms of this expansion, but the coefficients of higher powers of (1/v) have rather complicated expressions. Other useful results given by John (1966) include recursion formulas connecting V (yl, y2i ... , Yp) with Vv±2 (y1, y2, ... , Yp)

More recently, several authors have looked into the problem of computing multivariate t probabilities of the form

P = f f (x; v) dx,

(6.27)

A

where X has the central multivariate t distribution with correlation matrix R and A is any convex region. Somerville (1993a, 1993b, 1993c, 1994) developed the first known procedures for evaluating P in (6.27). Let MMT be the Cholesky decomposition of R (where M is a lower

triangular matrix) and set X = MW. Then W is multivariate t with correlation matrix Ip. If one further sets r2 = WTW, then F = r2/p has the well known F distribution with degrees of freedom p and v. Let A be the region bounded by p hyperplanes and described by

GW < d, where G = ( 9 1 , . , gp) and the jth hyperplane is g W = d1. For a random direction c, let r be the distance from the origin to the boundary


152

of A, that is, the smallest positive distance from the origin to the jth plane, j = 1,. .. , p. Then an unbiased estimate of the integral P in (6.27) is

Pr (F < r2/p)

(6.28)

.

To implement the procedure, Somerville chose successive random directions c and obtained corresponding estimates of (6.28). The value of P was then taken as the arithmetic mean of the individual estimates. Somerville (1997, 1998b) provided the following modification of the

above procedure. Let r* be the minimum distance from the origin to the boundary of A, that is, the smallest of the r for all random directions c. Divide A into two regions, the portion inside the hypersphere of radius

r* and centered at the origin, and the region outside. The probability content of the hypersphere is Pl

= Pr (F < r*2/p)

,

and this can be estimated as in Somerville (1993a, 1993b, 1993c, 1994).

If E(v) and e(v), respectively, denote the cdf and the pdf of v = 1/r (the reciprocal distance from the origin from and to the boundary of A), then the probability content of the outer region is 1/r*

P2

=

E( v)e(v)dv. fo

Since F = r2/p, the pdf of v is e(v)

=

vv-1

2vvl2r ((v + p)/2)

r (v/2) r (p/2)

(1 +

vv2)(v+r)/2

The strategy is to use some numerical method to estimate E(v) and then evaluate the integral P2 using the Gauss-Legendre quadrature. The approaches of Somerville (1997, 1998b) differ in that Somerville (1997) applied Monte Carlo techniques to estimate E(v) while Somerville (1998b) used a binning procedure. It should be noted, however, that an approach similar to these had been introduced earlier by Deak (1990). Somerville (1999a) provided an extension of the above methodologies

to evaluate P in (6.27) when A is an ellipsoidal region. This has potential applications in the field of reliability (in particular relating to the computation of the tolerance factor for multivariate normal populations) and to the calculation of probabilities for linear combinations of central and noncentral chi-squared and F. In the coordinate system of the transformed variables W, assume, without loss of generality, that


153

the axes of the ellipsoid are parallel to the coordinate axes and the ellip-

soid has the equation (w - u)TB-1(w - u) = 1, where B is a diagonal matrix with the ith element given by bi. If the ellipsoid contains the origin, then for each random direction c there is a unique distance r to the boundary. An unbiased estimate of P is then given by

Pr (F < r2/p) . If the ellipsoid does not contain the origin, then, for a random direction, a line from the origin in that direction either intersects the boundary of the ellipsoid at two points (say r > r*) or does not intersect it at all. If the line intersects the boundary, then an unbiased estimate of P is given by the difference

Pr (F < r2/p) - Pr (F < r2/p). If the line does not intersect the ellipsoid, an unbiased estimate is 0. As in the first procedure described above, this is repeated for successive random directions c, each providing an unbiased estimate. The value

of P is then taken as the arithmetic average. A modification of this procedure along the lines of Somerville (1997, 1998b) is described in Somerville (1999a). Somerville (1999b) provided an application of his methods for multiple

testing and comparisons by taking A in (6.27) to be

A = {xERP:maxcTx 0. The purpose is to calculate the value of q for arbitrary R and v and arbitrary sets B such that the probability content of A has a preassigned value ry. Somerville and Bretz (2001) have written two Fortran 90 programs (QBATCH4.FOR and QINTER4.FOR) and two SAS-IML programs (QBATCH4.SAS and QINTER4.SAS) for this purpose. QINTER4.FOR and QINTER4.SAS are interactive programs,

while the other two are batch programs. A compiled version of the Fortran 90 programs that should run on any PC with Windows 95 or later can be found at http://pegasus.cc.ucf.edu/-somervil/home.html

These programs implement the methodology described above to evaluate the probability content of A (A Fortran 90 programs MVI3.FOR used to evaluate multivariate t integrals over any convex region is described in


154

Somerville (1998a). An extended Fortran 90 programs MVELPS.FOR to evaluate multivariate t integrals over any ellipsoidal regions is described in Somerville (2001). The average running times for the latter program

range from 0.075 and 0.109 second for p = 2 and 3, respectively, to 0.379 and 0.843 second for p = 10 and 20, respectively.). The so-called "Brent's method," an interactive procedure described in Press (1986), is used to solve for the value of q. The time to estimate the q values (with a standard error of 0.01) using QINTER4 or QBATCH4 range from 10 seconds for Dunnett's multiple comparisons procedure to 52 seconds for Tukey's procedure, using a 486-33 processor. A problem that frequently arises in statistical analysis is to compute (6.27) when A is a rectangular region, that is,

P-

61

Zb2

by

f (x1i x2, ... , xp) dxp . . . dx2dxl. (6.29)

. . . al

ap

2

Wang and Kennedy (1997) employed numerical interval analysis to com-

pute P. The method is similar to the approaches of Corliss and Rall (1987) for univariate normal probabilities and Wang and Kennedy (1990)

for bivariate normal probabilities. The basic idea is to apply the multivariate Taylor expansion to the joint pdf f. Letting e3 = (aj + bj)/2, the Taylor expansion of f at the mid point (c1, C 2 ,- .. , cp) is

f (x1,x2,...,xp)

_

p

m-1 k=0

akf (Cl, C2, ... , cp) (x kl ! ... kp! axkl axkz ... axkp 1 2 p i=1 1

]k[

1

8mf (S t1, S2t,

]m[

t)

. ., Sp

E ml l ... mpl ax1 1 8x22 ... .

axmp r

- c)

kj

p

11 (x - cj )m' 7-1 , (6.30)

where L;j is contained in the integration region [aj, bj] and ]k[ denotes

all possible partitions of k into p parts. For example, in the case p = 3, ]2[ will result in 6 possible partitions of `2' into {k1, k2, k3}: {0, 0, 2}, {0,1,1}, {0, 2, 0}, {1, 0, 1}, 11, 1, 0}, and {2, 0, 0}. The main problem with computing (6.30) is the presence of high-order partial derivatives of f. Defining (f)k,k2...kp

=

ak,+k2+ +kp f

1

k1!k2!

k p!

kp k2 axk1, axe ...axp

(6.31)


155

Wang and Kennedy derived the following recursive formula

xTR-lx

(f

)k1k2...kP

-1

I \1 + k1

X 1

kP

k2

`... p2v(kl-ll)+11

1=01 2= 0

1

(f)1112...1P

P= 0

xTR-1 x1 V

J PC1 -tl,'2-t2....-Xp-1P

With regard to the last quadratic term, it should be noted that higher than second-order partial derivatives are all zero. T o carry out the computation of (6.31) for a given (k1i k2, ... , kr), one can

first let one lj be k1 - 1 (if this kj 54 1) and all the other l1's be their corresponding kg's;

next let l, and is be k,. - 1 and k9 - 1, respectively (if k,. # 1 and k, # 1), while all the other lj's take their corresponding k1's;

finally, let some lj be kj - 2 (if kj > 2) and all other la's be the corresponding kg's.

The total number of terms that contribute to computing (f )k1 k2...kp is at

most p(p+3)/2. Compared to the multivariate normal distribution, this number is larger (Wang and Kennedy, 1990). The following table gives the running times and the accuracy for computing (6.29) with v = 10.

Running time and accuracy for computing P in (6.29) p

10 9

8 7

6 5

Running time (min)

aj = -0.5 aj = -0.4 bj = 0.5

bj = 0.4

80 70 85 90 110 180

3 sig 10 sig

0 sig 3 sig 8 sig

aj = -0.3 aj = -0.2 bj = 0.3

bj = 0.2

2 sig 3 sig 5 sig 8 sig

4 sig 7 sig 10 sig

Another point to note about Wang and Kennedy's method is that when the integration region is near the origin it works better for larger v, while


156

when the integration region is off the origin it works better for smaller v.

The main problem with Wang and Kennedy's (1997) method is that the calculation times required are too large even for low accuracy results (see the table above). Genz and Bretz (1999) proposed a new method for computing (6.29) by transforming the p-variate integrand into a product of univariate integrands. The method is similar to the one used by Genz (1992) for the multivariate normal integral. Letting MMT be the Cholesky decomposition of R, define the following transformations P

Xi

Mj,kyk, k=1

Yj

=

Ui

Ui

I-1 y2k V + Ek=1

v+j-1

=

and

= dj + Wj (ej - dj) ,

Z3

where TT denotes the cdf of the univariate Student's t distribution with degrees of freedom T,

a

dj

=

(aaj),

ej

=

(bj)

=a

v+j -1 3

b=b

_1 y2'

V + ,k=1yk

v+j-1 1

V + Ek=1 yk2 1

a _

,

a.7 - Ek=1 Tnj,kYk mj,j


157

and

Applying the above transformations successively, Genz and Bretz reduced (6.29) to

P=

1

1

1

(el _ dl) I (e2 - d2) ... I (ep - dp) 0 1

J0

dw

(6.32)

f1

i

J

J

f (w) dw.

(6.33)

The transformation has the effect of flattening the surface of the original

function, and P becomes an integral of f (w) = (el - d1) ... (ep - dp) over the (p - 1)-dimensional unit hypercube. Hence, one has improved numerical tractability and (6.33) can be evaluated with different multidimensional numerical computation methods. Genz and Bretz considered three numerical algorithms for this: an acceptance-rejection sampling algorithm, a crude Monte Carlo algorithm, and a lattice rule algorithm. Acceptance-rejection sampling algorithm: Generate p-dimensional uniform random vectors w1i w2, ... , wN and estimate P by N

1

P = nr

h(My,), 1=1

where

h (x)

=

1

if aj < xj < bj, j = 1, 2.... , p,

0

otherwise

and

v + k-=1 yk 1

)

Tv+,7-1 (w1,j

L+j-1

2

,

j = 1,2,...,p, l = 1,2,...,N. A crude Monte Carlo algorithm: Generate (p-1)-dimensional uniform random vectors w1, w2i ... , wN and estimate P by

I' =

1

N f (WI)

AT l=1

an unbiased estimator of the integral (6.33).


158

A lattice rule algorithm (Joe, 1990; Sloan and Joe, 1994): Generate (p - 1)-dimensional uniform random vectors w1i w2, ... , wN and estimate P by 1

2(?z+wi1-1p lq

Here N is the simulation size, usually very small, q corresponds to the

fineness of the lattice, and z E RP-1 denotes a strategically chosen lattice vector. Braces around vectors indicate that each component has to be replaced by its fractional part. One possible choice of z follows the good lattice points; see, for example, Sloan and Joe (1994).

For all three algorithms - to control the simulated error - one may use the usual error estimate of the means. Perhaps the most intuitive one of the three is the acceptance-rejection method. However, Deak (1990) showed that, among various methods, it is the one with the worst efficiency. Genz and Bretz (2001) proposed the use of the lattice rule algorithm. Bretz et al. (2001) provided an application of this algorithm for multiple comparison procedures. The method of Genz and Bretz (1999) described above also includes an efficient evaluation of probabilities of the form b

P = J g(x)f(x)dx, a

where g(x) is some nuisance function. Fortran and SAS-IML codes to implement the method for p < 100 are available from the Web sites with URLs http://www.bioinf.uni-hannover.de/-betz/ and

and http://www.sci.wsu.edu/math/faculty/genz/homepage.

6.11 Probabilities of Linear Inequalities Let X be a random variable characterizing the "load," and let Y be a random variable determining the "strength" of a component. Then the probability that a system is "trouble-free" is Pr(Y > X). In a more complicated situation, the operation of the system may depend on a

6.11 Probabilities of Linear Inequalities

159

linear combination of random vectors, say ai X1 + a2 X2 + b, and the probability of a trouble-free operation will be

Pr(a1Xi+a2X2+b> 0),

(6.34)

where Xj are independent k3-dimensional random vectors, aj are kjdimensional constant vectors, and b is a scalar constant. Absusev and Kolegova (2001) studied the problem of constructing unbiased, maximum likelihood, and Bayesian estimators of the probability (6.34) when Xj is assumed to have the multivariate t distribution with mean vector µj and correlation matrix Rj. If X1 1, ... , X1n, and X 2 1 ,-- . , X2n2 are iid samples from the two multivariate t distributions, then - in the where case both µj and Rj are unknown - it was established that the unbiased and the maximum likelihood estimators are T Or (a1 X1 + a2T X2 + b > 0)

_

r (n1/2) r (n2/2) 7rr ((n1 - 1)/2) r ((n2 - 1)/2) 2

x f 1j=1 (1 - v )(n3)/2 dvldv2 2

and ai Xn, + a2 Xn2 + b

Pr(a1X1+a2X2+b>0)

al Snl+l al + a2Sn2+1a2

respectively, where c1

=

vj? 0 j=1

j=1 ni 1

= _E ni Xm' m=1

ni+1

(nj + 1) xj

=

1: xjm, M=1

ni+1

(nj + 1) Sni+l

(Xjm - Xj) (Xj.m -;)T ,

= m=1


160

and x,,,+i = x. A Bayesian estimator of (6.34) with unknown parameters µj and Rj and the Lebesgue measure p(O)dO = dµdR was calculated to be

F ((nj - kj)/2)

2

PrB (aTX1 + a2 X2 + b > 0)

11 j_1 7rF ((nj - 1)/2) (nj + 1)k' 2

X

F

((nj

-k -

nJ

2

x

1 )/ 2 )

j

11 F ((nj - 2kj - 1)/2) j=1 g

[J (1 -

J 2 j=1

Z3?)

2

dzldz2i

where 2

Q2

=

z? < 1, j = 1, 2, E zj

2

nja

j=1

R. + b > 0 j=1

This Bayesian estimator is biased and is related to the unbiased estimator via the relation PrB (aTX1 + a2 X2 + b > 0)

= APr (aTX1 + a2 X2 + b > 0) ,

where

A = 71 F ((nj - kj - 1)/2) F ((nj - k)/2) (nj + 1)k j=1

F (nj - 2kj - 1)/2) F (nj/2)

nk)-

The coefficient A can be expanded as k A = 1+n-nkk+Ol I) n2

I

where n = max(nl, n2) and k = max(kl, k2). `Therefore, the Bayesian estimator is asymptotically unbiased as n -> oo. Substantial literature is now available on problems concerning probabilities of the form (6.34) for various distributions. For a comprehensive and up-to-date summary, the reader is referred to Kotz et al. (2003).

6.12 Maximum Probability Content Let X be a bivariate random vector with the joint pdf of the form

f (x)

= g ((x - µ)T R-1 (x - µ))

,

(6.35)

6.13 Monte Carlo Evaluation

161

which, of course, includes the bivariate t pdf. Consider the class of rectangles

=

R(a)

{(x1,x2) : lxii < a, 1x21 < A/(4a)}

with the area equal to A. Kunte and Rattihalli (1984) studied the problem of characterizing the region R in this class for which the probability P(R(a)) = Pr(X E R(a)) is maximum. As noted in Rattihalli (1981), the characterizations of such regions is useful for obtaining Bayes regional estimators when (i) the decision space is the class of rectangular regions and (ii) the loss function is a linear combination of the area of the region and the indicator of the noncoverage of the region. It was shown that, for any fixed ) > 0, the maximal set is

{(x1ix2):Ixl-All 0 for all j. Then, since the integral over X covers the region X < Xo, the integral over Z is determined by the inequality BZ < Xo, and the bounds can be written as zp

I E {Fj(x)}

Fj(x)

(7.3)

j=1

j=1

(where Fj denotes a cdf), one can now bound G by xjs + bjz

HE HE

G (xl s, x2s, ... , xps)

b

P

_

Pr

1-VYj-bjYo<xjs} J2

j=1 P

11 D (xj8) j=1

Substituting this result into (7.1) and applying (7.3) once more, one obtains the lower bound for P given by Dunnett and Sobel (1955) as

P>

T0O p

11 iP (xjs) f (s)ds j=1 P

f00

> 11 J j=1

(xjs) f (s) ds °°

7.1 Dunnett and Sobel's Probability Inequalities

167

P

lPr{Zj <xjs}

j=1 P

11 Pr {Xj < xj }

(7.4)

.

j=1

This lower bound for P holds more generally for any correlation matrix R with rij > 0 and any arbitrarily fixed.. (x1, . , xp). This is a conse-

quence of the fact that P is an increasing function of each rij for all i # j, while other correlations are held fixed. It can be shown further that P

Pr (X1 > x1 i ... , Xp > xp)

> H Pr (Xj > xj) j=1

and P

Pr (IX1 I < x1, ... , I XpI > xp)

11 Pr(IXjI>xj) j=1

Since the bound (7.4) does not depend on rij, it can be calculated easily from a table of the cdf of the univariate Student's t distribution. Dunnett and Sobel (1955) also obtained two sharper bounds by slight modifications of the above arguments: For even p > 2,

P>

p/2

11 Pr {X2j_1 < x2j_1, X2j < x2j} ,

(7.5)

j=1

and for odd p > 3,

P>

(p-1)/2

Pr {X1 < x1 } H Pr {X2j < x2j, X2j+1 < x2j+1 }

.

j=1

(7.6)

In the case where rij = p for all i 54 j and xj = x for all j, inequalities that are sharper than (7.5) and (7.6) can be obtained. Let 01(p)

= Pr(Xi oo. Recently Seneta (1993) pointed out that the "sub-Markov" inequality /31(p)

>

[Pr {Xi < x, X2 < x}]p-1 / [Pr {X1 < x}]p-2

(7.9)

is sharper than the corresponding inequality /31(p)

>

[Pr {X1 < x, X2 < x}]p12

(7.10)

as given by (7.8). This fact is illustrated in the following table, which is taken from Seneta (1993).

Comparison of the bounds (7.10) and (7.9) for P. x chosen such that the true value of P = 0.95

p=3

V

x

10

2.34 2.24

15

20

60

p=9

2.19 2.10

10

2.81

15

2.67 2.60 2.48

20 60

Bound (7.10)

Bound (7.9)

0.945

0.946

0.946 0.945 0.944

0.947 0.946 0.945

0.921 0.924

0.921

0.926 0.934

0.927 0.936

0.924

7.2 Dunn's Probability Inequalities

169

Actually, (7.9) is a particular case of the following inequalities /31(p)

> Pr(Xj <x;j=1,...,m-1) x {Pr (Xm < x I Xj < x; j = 1, ... , m - 1)}r-'"+1 (7.11)

and

/32(P)

>

Pr(lXjl <x;j = 1,...,m-1) x {Pr (IXmI < x I 1Xjj < x; j = 1, ... , m - 1)}P_m+1

given by Glaz and Johnson (1984), who also provided a formal proof of the fact that (7.11) is sharper than (7.8). Dunnett (1989) wrote a Fortran programs for evaluating the integral (7.2). It uses Simpson's rule to compute an approximation to (7.2) in such a way that a prescribed accuracy is achieved. To approximate the integral of a function a(z), say, over an interval [a, b] using Simpson's rule, the value of the function is computed at the two end points and at the midpoint of the interval; then the approximate value of the integral is given by

{a(a)+4a1 a2

bounded

b

I +a(b)}

b

6

a,

by a4 (a, b) (b/-a) 5 /28)80, where a4 (a, b) is a bound with its error on the absolute value of the fourth derivative of a(z) over the interval (see, for example, page 66 in Shampine and Allen, 1973). The central processor unit time (on a VAX 8600 computer using single-precision arithmetic) taken to compute (7.2) ranges from 0.01 to 2.37 seconds for cases of equal correlation (r2j = p) and identical ranges of integration (xj = x). Slightly longer computing times are required for unequal correlations or different limits of integration. Dunnett (1989) suggested that his program can be used along with an appropriate numerical integration routine, such as the Integral Mathematical and Statistical Libraries'

(1987, Volume 1, Chapter 4) QDAGI, to evaluate the multivariate t probability integral (7.1).

7.2 Dunn's Probability Inequalities The univariate Student's t distribution has the property that the probability evaluated from -x to +x is an increasing function of the degrees of freedom v - this also applies to the probability from -oo to +x (see,

Probability Inequalities

170

for example, Ghosh, 1973, for details). Dunn (1965) pointed out that this monotonicity does not generalize to p dimensions in the usual multivariate t distribution. Specifically, let X have the central p-variate t distribution with degrees of freedom v and correlation matrix R. If F(x) is defined by

F(x)

= Pr {f1P=1 - x < Xk < X},

then F(x) equals the probability mass in the multivariate t distribution evaluated over a p-dimensional hypercube centered at the origin of the half side x and F(x) is the distribution of the maximum of the absolute values of the p X variables. Similarly, if G(x) is defined by G(x)

= Pr {f1P=1 - oo < Xk < x} ,

then G(x) equals the probability mass evaluated from -oo to x in each direction and G(x) is the distribution of the maximum of the p X variables. Dunn showed that, for any given x > 0 and degrees of freedom v1 > v2i there exists an integer K such that, for all p > K, Fp,v, (x)

1 be independent samples from a bivariate normal distribution with zero means, variances o , 0'2, and covariances 0`1a2pi, pi 1< 1. Let Yi1, i = r + 1, ... , r + n and Yi2, i = r + 1, ... , r + m be independent normal samples with zero means and variances or1 and Q2, respectively, and I

7.4 Siddk's Probability Inequalities independent of (Yil, Y i 2 ) ,

171

i = 0 , 1 , 2, ... , r. Define

(X1, X2) =Y10 - ,Y20 - S2), S1 where r+n

EY2il

1

Si

=

r+

i-1

and

r+m

S2 =

2

r

+ m

Yi2 i-1

Halperin (1967) then showed that the probability integral of (X1, X2) satisfies the inequality

Pr(jXiI <x1,1X2j<x2) > Pr(IXiI <x1)Pr(jX2j <x2) for all real numbers xl and x2.

7.4 Sidak's Probability Inequalities In the bivariate case considered above, it is assumed that the correlation between Yil and Yi2 may be different for different i's. For a general p, but for a special correlation structure of Y's, Sidak (1967) established the following result. Let YT = (Y1, Y2, ... , Yp) have a p-variate normal

distribution with zero means and an arbitrary correlation matrix. Let ZT = (Zi1, Zit, ... , Zn,), i = 1, ... , n be a p-variate normal random sample, which is mutually independent and independent of Y, each of which has zero means, unit variances, and the decomposable correlation structure given by Corr (Zki, Zkj)

=

bibs

for i, j = 1,...,p; i # j; k = 1,...,n with 0 < bj 2. Sidak (1971) went on to show that, if I

Corr (Yi, Y-)

=

cicjrid

for i, j = 1,2,...,p; i Oj with I cj < 1 (j = 1,2,...,k) and {r2j} any fixed correlation matrix, and if Corr (Zii, Zip)

=

bciblj I

b1 1 b. In an earlier paper, Tong (1970) obtained similar results for a much larger class of random vectors.

7.5 Tong's Probability Inequalities We noted in Chapter 1 that the multivariate t density is Schur-concave in the particular case rid = p, i j. Tong (1982) used this property to derive certain probability inequalities. He showed that if f : RP -> [0, oo) is Borel-measureable and Schur-concave, then, provided that the integral exists, fA(X) f(y)dy is also a Schur-concave function of (x1,. .. , xp),

7.5 Tong's Probability Inequalities

where A(x) denotes the rectangular set

A(x) = {YIYER',IyjI <xj,i=1,...,p}. Taking f to be the pdf of the multivariate t, it follows that

Pr(IX;I < xj,j = 1,...,p) < Pr(jXjj < -t, j = 1,...,p),

where= -(xl++xv).

173

8

Percentage Points

From the 1950s numerous authors have tried to compute the percentage points of multivariate t distributions. It is an indication of the interest in problems leading to applications of this "new" distribution. This research continued well into the 1990s and is still going strong. Although some of the results have by now lost their practical importance - in our opinion - it is essential for historical reasons to describe a majority of these contributions. This will certainly assist historians and experts in multivariate distributions to gain a better perspective of the developments in the area. Moreover, many of the techniques involved in these calculations are ingenious and worthy of emulation and further investigation.

8.1 Dunnett and Sobel's Percentage Points Let (X1, X2) have the bivariate t distribution with joint pdf (6.1). For given v, p and probability level ry, let d denote the equicoordinate percentage point satisfying f (xl, x2; v, P)dxtdx2

=

-Y-

The value of d can be determined for any v by trial and error using (6.3) and (6.4). However, this procedure becomes more involved as v increases. Dunnett and Sobel (1954) derived an asymptotic expansion in powers of 1/v that expresses d in terms of the corresponding quantity e for the bivariate normal distribution: e is defined by e

re

f (xl, x2; oo, p) dxldx2 174

=

'Y

8.2 Krishnaiah and Armitage's Percentage Points

175

and can be obtained by interpolation in the classical tables of Pearson (1931), for example. Their expansion yields a good approximation of d even for moderately small values of v. The method of derivation is essentially the same as that used by Fisher (1941) in deriving an asymptotic expansion for the percentage points of the univariate Student's t distribution. Up to the terms of O(1/v), Dunnett and Sobel obtained

d = e-4v

e2+1-t't)

(t)

and by inverting

d+4 Jd2+1ll

=

e

,d) I

()

.

They also tabulated numerical values of the coefficients of the terms 1/vi for i = 0, 1, 2,3,4.

8.2 Krishnaiah and Armitage's Percentage Points Consider X having a central p-variate t distribution with degrees of freedom v and with the equicorrelation structure rid = p, i :/ j. Krishnaiah and Armitage (1966) evaluated the multivariate percentage point d by solving the integral P(d)

=f

d

...

fd

J

.f (xi, ... , xr; v, p)dxi ... dxp = 7

(8.1)

00

and produced extensive tables for all combinations of p = 1(1)10; v = 5(1)35; p = 0.05(0.05)0.9 and y = 0.90, 0.95, 0.975, 0.99 (see also Armitage and Krishnaiah, 1965). These computations use the approximation that xv-1

P(d)

(-x2) F exp (u/2)

2 foC

x

2

fo0

y

exp -y2)

v(1-P) dx + V 1-p

dydx,

is the cdf of the standard normal distribution and the upper limit c is chosen large enough to make the error of approximation as small as desired. Krishnaiah and Armitage (1966) took c = 10. where

Percentage Points

176

8.3 Gupta et al.'s Percentage Points Gupta et al. (1985) solved equation (8.1) by setting

IP(d) -

71

10-s

dl)

=

1 -'y;

this approximation had been suggested previously by Dunn (1958, 1961). By Bonferroni's inequalities, one notes that

1-ly-El(y,p) d2) . i<j

Thus, if cl (ry, p) is sufficiently small, then one can use dl as a good estimate. A modified second approximation is the value d2 satisfying 2p Pr (XZ > d2)

=

1 - _Y + El ('y, p)

.

This time, one notes that -ez (,y, p) d22 iX? >d2)-EI('Y,p) 2 i<j

and e3

(Y,p)

Pr (Xi > d2,Xj2 > d2,Xk > d2) .

_ i<j 0, the absolute value of 'y - P(d2)

may be expected to be sufficiently small for the tail of the p-variate t distribution to correspond to 1- ry for values of 7 > 0.95. For the particular case p = 2, µi = µz = 0, and the equicorrelation structure rij = p, i # j, Siotani (1964) tabulated estimates of the probability in (8.3) for all combinations of d = 2.0(0.5)4.5; v = 10(2)50(5)90,100,120,150, 200, oo; and I p 1= 0.0(0.1)0.9, 0.95. He also illustrated applications to interval estimation of the parameters in the model of a randomized block design and for coefficients in a normal regression equation.

8.7 Graybill and Bowden's Percentage Points Graybill and Bowden (1967) derived bounds for d satisfying (8.3) for the special case p = 2 and p = 0. In this special case (8.3) becomes

Pr{X1 2, J1 = -t(e1), and Jo = 1 (see also Kwong, 2001a). There are three commonly known approaches for determining the percentage points d 1 , . . . , dp in (8.9) (after setting P = ry): the step-up procedure, the step-down procedure, and the simulation approach (Dunnett

8.13 Dunnett and Tamhane's Percentage Points

185

and Tamhane, 1995). We shall describe them below by means of recur< X(p) denote the sive algorithms. Throughout we shall let X(1) < order statistics of X1, . . . , X p and let c 1 , . . . , cp denote the corresponding ordering of d1,...,dp. Step-up procedure

(i) Take cl to be the 100'y percentage point of the univariate Student's t distribution with degrees of freedom v. (ii) Solve the equation Pr (X(l) < cl, X(2) < c2) Pr (Xl < c1i X2 < c2) + Pr (cl < X1 < c2i X2 < cl ) ry

for c2 by evaluating the two bivariate probabilities using (8.9) and the value for cl defined in (i). (iii) Solve the equation Pr (X(1) < cl, X(2) < c2i X(3) < c3)

= Pr (XI < c1i X2 < c2, X3 < c3) + Pr (Xl < c1, c2 < X2 < c3, X3 < c2) + Pr (cl < X1 < c2i X2 < c1, X3 < C3)

+ Pr (c1 < Xl < c2i c1 < X2 < c3, X3 < cl )

+Pr(c2 <X1 20 using tabulated values for p = 20.

Freeman and Kuzmack (1972) for p = 6, 8, 10(5)30; No = 10, 20, mean (40-70), 50, 100, mean (90-500), 500; p = 0; and ry = 0.90, 0.95, 0.99, where v = p(No - 1). Gupta et al. (1973) for p = 1(1)10(2)50; v = oo; p = 0.1, 0.125, 0.2, 1/3, 0.375, 0.4, 1/2, 0.64, 0.625, 2/3, 0.7, 0.75, 0.8, 0.875, 0.9; and y = 0.75, 0.9, 0.95, 0.975, 0.99.

Amos (1978) for p = 100; v = 100; p = 0.01; and ry = 0.05, 0.1, ..., 0.95.

Ahner and Passing (1983) for 1 < p< 20; 2 < v < 120, v = oo; p = 0; and ry = 0.95, 0.99.

Bechhofer and Dunnett (1988) for the most comprehensive table to date for p = 2(1)16, 18, 20; v = 2(1)30(5)50, 60(20)120, 200, oo; p = 0.0(0.1)0.9, 1/(1 + f); and ry = 0.80, 0.90, 0.95, 0.99. Kwong and Iglewicz (1996) for p = 4, 5; v = p(1)20, 24, 30(10)100,

120, oo; p = -1/(p - 1); and ry = 0.90, 0.95, 0.99 (note that the correlation matrix is singular).

There has been relatively little work concerned with the percentage points of multivariate t distributions when the correlations are not equicor-

related. Apart from those mentioned above, four results known to us are

In the case that the (i,j)th element of the inverse of R is

_

ria

2p/(p+ 1),

if i = j, -2/(p+ 1), if i i4j,

Goldberg and Levine (1946) computed the percentage point d satisfying d

00

rd

J-00

f (x1i... , xP; v) dxl ... dxr =

'Y

(8.13)

for combinations of p = 3; v = 1(1)30(3)60(15)120, 150, 300, 600, oo;

189

8.15 Other Results

and ry = 0.50, 0.75, 0.90, 0.95. This seems to be the earliest paper on this topic. In the case 2min(i,j){1-max(i,j)1

r2 =

p

l

1'

Freeman et al. (1967) computed the percentage point d satisfying (8.13) for combinations of p = 3(1)5; (v/p) + 1 = 10(10)100, 200, 500; and ry = 0.95. See also Bechhofer et al. (1954) and Table 4 in Dunnett and Sobel (1954).

In the case that the (i,j)th element of R is nine rz7

nj)'

(no + ni)(no +

where nk denotes some treatment sample size, Dutt et al. (1976) computed d in (8.13) for all combinations of y = 0.95, 0.99 and 3 < ni < 12, i = 0, 1, 2, 3 with p = 3 and the degrees of freedom v = E(ni - 1). See also Dutt et al. (1975). In the case rig = 0 for all i 54 j except that ni + ni+1

ni+1 -+n . + 2

for some treatment sample sizes nk, Lee and Spurrier (1995) provided

tables of one-sided and two-sided percentage points - of the form (8.13) - for 3 < p < 6 and nk = n (the balanced case). Liu et al. (2000) extended these tables for 3 < p < 10; v = 5(1)8(2)20, 25, 30, 40, 60, 120, oo; and ry = 0.90, 0.95, 0.99. See also Somerville et al. (2001).

Calculations of percentage points for singular correlation structures of the form rig = -bib; are discussed in Spurrier and Isham (1985) and Kwong and Iglewicz (1996). The former provided tabulations of the

percentage points for p = 3, 3 < nl k,

Sampling Distributions

194

_

E (IAIk)

IF (v/2 - kp) rp ((n - 1)/2 + k) IRIk v-kpr (v/2)

r((-1)/2)

v>k, _ E (IAIk A)

((n - 1)/2 + k)r (v/2 - kp - 1)

X

v-(l+kp)r (v/2) rp ((n - 1)/2 + k) IRIk R,

rp((n-1)/2)

n + 2k > 1,

217 (v/2 - kp + 1) Fp ((n - 1)/2 + k)

E (IAIk A-1)

E [(trA)2]

v > 2(kp + 1), IRIk R-1,

Ill-kp(n + 2k - p)I' (v/2) rp ((n - 1)/2) n + 2k > p + 2, v > 2(kp - 1),

_

(1

- 2/v) (1 - 4/v) v>4

[(n - 1) (trR)2 + 2tr (R2)]

,

and

=

E [tr (A2)]

n-1 [ntr (R2) + (trR)2] (1 - 2/v) (1 - 4/v)

,

v>4, where k is any real number and v > 0. These expectations are important tools in developing estimation theories for the correlation matrix, inverted correlation matrix, trace of the correlation matrix, and other characteristics of the correlation matrix, of the multivariate t model under quadratic loss functions. Extensions of these expectations to the class of scale mixtures of normal distributions - which may be useful in inferential works having a t distribution or the scale mixture of normal distributions as the parent population - are discussed below. Sutradhar and Ali (1989) derived an elementwise expression for the variance-covariance matrix of A. Letting mij denote the (i, j)th element of R'/2, they showed that Cov (Aij, Akt)

=

v2 _n

3(n - 1) E miumjumkumlu u=1

+(n - 1)(n - 2) EPmiumjumkumlu u=1

+(n - 1)2

miumjumkumlu u 0, n > 1 +p, and rp(-) is as defined in (9.6). Some expectations of A useful for estimating R are IR"Ir

2pr

r rn(n l )r)

E (JAI') E [IAA

k A]

=

2

kp

(n + 2k)

rp (n/2 + k)

r (n/2)

Y2pr,

k

IRS R'Y2kp+2,

and

E [(trA)2]

=

nry4

[n (trR)2 + 2tr (R2)]

,

9.1 Wishart Matrix

197

where 'y E R, k E R, n + 2k > 0, 'y2pr = E(7- 2pr) > 0, 'Y2kp+2 = E(T2kp+2) > 0, and ry4 = E(T4) (all assumed to exist).

The pdfs of A in the real and complex cases - under the independence model (9.1) - were originally studied by Cornish (1955) and Gupta (1964), respectively. Nagarsenker (1975) provided a very detailed study

of the distribution of A and its quadratic forms. He investigated both the noncentral real and the noncentral complex cases. Let Y be a p x n matrix of iid normal random variables with means E(Yij) = pij and covariance matrix a2R. Assume that S is an indepen-

distribution. Then the

dent random variable having the noncentral version of A is defined by

=

aij

l

nn`

(Yjk-Yj),

Q2E(Yik-Yz) k=1

where 1

Y

n

n 1Yj. j-1

In the real case, S2A has the noncentral Wishart distribution. In the complex case, it should be interpreted as a Hermitian positive definite matrix having the noncentral complex Wishart distribution (James, 1964). In the noncentral real case, Nagarsenker (1975) established that

the pdf of A is given by the complicated expression involving zonal polynomials

f(A) (aF)p(n-1) I R I-(n-1)/2I A I(n-p)/2

exp {-trR-1/z, T/ (2a2) } (2v)P(n-1)/2rp ((n - 1)/2) r(v)

Up(n-1) (or )2k F (v

0

+ k + p(n - 1)/2) Ck (R-lµ,LT R-1A) k!((n - 1)/2),, (4vQ4)k {1 + (Q'2trR 1A) /2vQ2}

X

k=O k

+p(n2

1)

+ k,

(9.8)

where ,={kl,...,km}, k1>k2>...>k,,,,>0,k1+k2+ (x)

r(r-1)/4

-

+ km = k,

rP (X, K) )

-

1

...

(x+r -

2


198

and Ck(T) are symmetric homogeneous polynomials of degree k in the

latent roots of T. In the particular case a = a' and it = 0, (9.8) reduces to the expression given in Cornish (1955). If B is an (n - 1) x (n - 1) symmetric positive definite matrix of full rank, then Nagarsenker (1975) established further that the quadratic form Q = ABA' has the formidable pdf

f(Q) (0.l)P(n-1)

I R I-(n-1)/2I

Q

I(n-p)/2

cP(n-1) (2v)P(n-1)/2rp ((n - 1)/2) r(v) I B JP/2 X 00

C,, (-R-1Q/2v) C, (B-1) r (p(n - 1)

k=O c

k! (Q/Q')2k Ck (In-1)

+k+v

2 (9.9)

In the particular case o, = ci' and B = In-1, (9.9) reduces to equation (14) in Cornish (1955). Nagarsenker also provided the joint cdf and the moment generating function of Q as well as the corresponding expressions for the noncentral complex case (which generalizes those given in Gupta, 1964).

9.2 Multivariate t Statistic

A random variable X with iid copies X,, X2, ... is said to be in the domain of attraction of the normal law if there exists an -4 oo such that n

-I: 1

-+

an i=,

N(0,1)

as n -4 oo. It is well known that, for X in the domain of attraction of the normal law, the t statistic defined by _A) E 1 (Xi Tn == -+ N(0,1) (9.10) En

I (Xi -

X)2

Ei

as n -> oo, where, as usual, (1/n) 1 Xi. Sepanski (1994, 1996) provided two multivariate analogs f (9.10). Let

X be a p-variate random vector with mean vector µ and covariance matrix E. Also let X1, X2, ... be iid copies of X. Then X is said to be in the domain of attraction of a p-variate normal law if there exists

9.3 Hotelling's T2 Statistic

199

an -+ oo such that

1 a

(X i - N-)

-* N (0, C)

(9.11)

n i=1

for some nonsingular matrix C. Sepanski (1994) defined the multivariate

t statistic by n

Tn = Dn 1/2

(Xi - µ-)

,

(9.12)

i=1

where, for some sequence bn > 0, Dn = Cn + bnI and n

Cn =

(Xi - X) (Xi - X)T. i=1

Note that Cn is symmetric nonnegative definite while Dn is symmetric positive definite. Under the assumption that X satisfies (9.11), Sepanski (1994) showed that Tn -> N(0, I) as n -4 oo. Sepanski (1996) established the same limiting result under weaker conditions by taking n

Tn = Cn1/2 E (Xi - µ)

(9.13)

i=1

and considering its behavior when X is in the generalized domain of attraction of a normal law, which means that there exist matrices An and vectors µn such that n

Xi - µn -4 N (0, I)

An

(9.14)

i=1

as n -* oo. See Hahn and Klass (1980a, 1980b) for several examples of random vectors satisfying this condition and for an algorithm for constructing the normalizing matrices A.

9.3 Hotelling's T2 Statistic A customary approach to the estimation/testing problem is based on the so-called Hotelling's T2 statistic. It is defined by H,

= n2 (X - µ)T Cn 1 (X - µ)

Under the normality, it is well known that (n - p)Hn/(p(n - 1)) is distributed as an F distribution with degrees of freedom p and n - p (see, for example, Anderson, 1984). The distribution of H,2, has been studied


200

under a mixture of two normal distributions by Srivastava and Awan (1982) and Kabe and Gupta (1990). Iwashita (1997) investigated the asymptotics of Hn under an elliptical distribution. Unfortunately, there are no direct results for the specific case of multivariate t distributions. For completeness, however, we shall survey the results when X has an elliptical distribution. In this case, the characteristic function of X can be written as

0(t) = exp {itT m} I (mTQ-lm)

(9.15)

(Kelker, 1970) and the parameter

for some nonnegative function

11

'F

_

(0) {W'(0)}2

- 1,

which controls the kurtosis of the distribution. Iwashita (1997) provided an asymptotic distribution of H.2 under the null hypothesis that m = µ and a local alternative of it. Up to the order of 1/n, the asymptotic null pdf of Hn is given by 2

f (x)

=

gp(x) +

E ojgp+2j (x) j=0

+0

(n)'

where CO

-Ip{p+k(p+2)},

=

-2p{1 -r.(p+2)1,

Cl

=

c2

I

p(p + 2) (1 - n),

and denotes the pdf of a chi-squared distribution with degrees of freedom k. Iwashita (1997) also derived the percentiles and approximate powers of the Hn statistic. An asymptotic expansion of the cdf of Hn under the two assumptions (i)

E(II Y 118) < oo, where Y = E-112(X - µ) and X is a p x 1

random vector with mean vector µ and covariance matrix E; (ii) the distribution of Y = (Y1, ... , Yp) has an absolutely continuous component with a positive density on some nonempty open set U such that 1, y1, . on U

.

.

, yp, yi , YlY2, ... , yep are linearly independent


201

is given by Fujikoshi (1997). It takes the form 3

= Gp(x) + 1 E NjGp+2j (x) + o I 1n

Pr (H,2, < x)

I

(9.16)

j=0

uniformly for all positive real numbers x, where Gk(.) denotes the cdf of a chi-squared distribution with degrees of freedom k. The coefficients /3j's are given by 6(r( 1)) 2

_ 4p2 +

=

00

1 _ 2p

N1

K41) 4

3

3((1))2 + 2 k4(1)

1

1

2

2

02

= 4p(p+2) - 2 (K32))

__4

X4

1)'

and

(,(2) )2' /33

3

3 (x(1))2 +2

3

where K31)

= J: (X(Zj,k))2, i,j,k

K(2)

K(i,i,k)/£(j,k,k)'

i,j,k X41)

=

K(i,i,j,j)

i,j

are the jth cumulants of Y. If X has the elliptical distriand bution given by (9.15), then a3 vanishes to zero and /3k reduces to ck for k = 0, 1, 2. Kano (1995) obtained the same asymptotic expansion as in (9.16), using a different method. It is well known that, for large samples, H,2, has a limiting chi-squared distribution with degrees of freedom p. The usual underlying assumption for this result is simply that E 11 X j1< oo. More general limiting behavior of H,2, has been studied by Eaton and Efron (1970), Sepanski (1994), and Fujikoshi (1997). Sepanski (1994) showed that, under the


202

assumption of the generalized domain of attraction (defined in (9.14)), the modified Hotelling's T2 statistic Hn2

= n2 (X - µ)T Dnl (X - µ)

still has an asymptotic chi-squared distribution with degrees of freedom p. Eaton and Efron (1970) studied the distribution Hn when X has

orthant symmetry, that is, X has the same distribution as DX for any choice of the diagonal matrix D with diagonal elements equal to ±1. We shall now consider the Hotelling's T2 statistic in the context of testing equality of means. Suppose X1 = (X11, ... , X lnl )T and X2 = (X21, ... , X2n2)T are two samples of size n1 and n2, respectively. In analogy with (9.2), assume that X1 and X2 have the joint pdf given by f (x1, x2)

a

JRI

-n/2

-(v+np)/2

n:

2

v-2+E

(xij - µi)T R1 (xij - µi)

i=1 j=1

where n = n1 + n2. It is immediate that Xij is p-variate t with mean vector µi, correlation matrix R, and degrees of freedom v. Also, the elements of the combined sample of size n = n1 + n2 are pairwise uncorrelated. The Hotelling's T2 for testing equality of means takes the form (X1 T2

nn+n2 (X1

- X2)T spooled

- X2)

,

where n,

1

Spooled

=

2

nl +n2 - 2

E z=1 j=1

(Xi.7 - Xi)

T T.

Sutradhar (1990) derived the nonnull distribution of the T2 statistic, given by the pdf f (t2)

r v 2 2

I #6/(,-2) (r + 1, 2- 1)

nl+n2-p-1 XOf r+ p 2 2' where

I'(k)r(m)xk-1 r(k + m)(1 + x)k+m

'

(9.17)


203

and

nnin2 n2

R-1 (i1 - P2)

(µ1 -

Note that, under Ho : it, = µ2, where S = 0, the pdf of T2 in (9.17) reduces to

f(t2)

= of

n1+n2-p-1 (2)

2

which implies that, under Ho, T2(n1 + n2 - p - 1)/p has the usual F distribution with degrees of freedom p and n1 + n2 - p -1. Thus the null distribution remains the same as in the normal case. Furthermore, the power of the Hotelling's T2 test can be computed by using the nonnull pdf in (9.17). Kozumi (1994) considered testing equality of means when the two samples X1 and X2 have mutually independent t distributions with equal correlation matrices and equal degrees of freedom. When the sample sizes are equal (say, n1 = n2 = n) the T2 statistic is given by Td = nyTShcy,

where y and Sd are, respectively, the sample mean and the sample covariance matrix of the differences yj = xlj - x2j. For unequal sample sizes, assuming without loss of generality that n1 < n2i the T2 statistic is given by

TS = n12TS312, where nl Zj

=

Xlj

1 X 2j j +

V

n2

/-n -1n-2 1

n2

X2s -

n2 1

1

X2s 1

and z and S8 are the sample mean and the sample covariance matrix of the zj's. It should be noted that Ts reduces to Td in the case n1 = n2 = n. Under the Ho : µ1 = z2, (n1 - p)TS /(p(n1 - 1)) has the usual F distribution with degrees of freedom p and n1 - p. The nonnull pdf of T3 is given by the infinite sum involving Student's t pdfs

f (t8)

(n1V ( 1) F2

x

(v/2) 2 8

-

k!B (p/2 + k,((n1

p/2+k-1

n1-1

xJ(x,n1,n2,k,6,v),

t2 1

+

8

n1-1

v p)/2) -(ni/2+k)


204

where 6 = (µ1

- µ2)T R-1(µl - 12) and the integral 1

J(x;nlin2,k,6,v)

L

{x(1 - x)}k+v/2-1 {n26x(l - x)

-(k+v)

n2 ) x+vn2

+V

ni

\\

dx.

ni

Kozumi (1994) also provided an expression for the cdf of Ts and calculated the powers of TS corresponding to the sizes a = 0.01, 0.05, p = 5 and for various values of n1i n2, 6, and v.

9.4 Entropy and Kullback-Leibler Number The forms of entropy and Kullback-Leibler number for the multivariate t distribution were discussed earlier in Chapter 1 (see equations (1.27), (1.29), and (1.31)). Here, we shall discuss the corresponding sampling properties. The entropy for the central p-variate t involves the correlation matrix

R, and it is known that the maximum likelihood estimator of R for a sample of n observations is based on the Wishart matrix A in (9.4). Hence it is of interest to consider the sampling properties of the difference 6 = H(X; A) - H(X; R). Guerrero-Cusumano (1996a) derived the corresponding moment generating function, mean, variance, and some asymptotics. Specifically,

M(t) =

vPt/2rn-l+t)r(v-tp) t\

2

1

2

P

n -i

2

rP(n-'-)rrl 2

(vl

\2

p21+21ogv'

2

i=1

Var (6)

4 i-1

2

and

E 2

v S*

as n -a oo, where

=

n

-4

N(0, p)

2

v [2, - p log {v(p - v) j]

denotes the digamma function.

Note that

9.4 Entropy and Kullback-Leibler Number

205

Hu(X; R) = H(X; A) - E(5) is an unbiased estimator for H(X; R) with E(H,,,) = 0 and Var(Hv,) = Var(8). Also note that, as v - oo, (n_2

E(8)

2E z-1

2) +plog2,

coinciding with the result given in Ahmed and Gokhale (1989) for the multivariate normal distribution. The expression for Var(S) given above is also valid for the multivariate normal distribution since it is independent of v. The Kullback-Leibler number for the central p-variate t is given by (1.31). The corresponding maximum likelihood estimator for a sample of n observations is

/,/-2) nAll-2log(vT

T(X;R) = S2--log

the sampling quantity of interest is the difference S = T(X; R) T(X; R). Guerrero-Cusumano (1996b, 1998) derived the corresponding moment generating function, cumulant generating function, cumulants, mean, variance, and some asymptotics. Specifically,

Mt O = (J) Ptr2rP1(n-1+t)r(v+tpl \ 2 J 2 J FP(n21)I'\2/ K(t)

2tlog(v-2)+logr(v 2pt)-logr\21 +E{logrn-2 i-t)

l

K3

pj-"bu-)

=2 E(S) =

Var (6)

p)

(v

2

PV)(1)

{

J' i)

2

2

2\v21

4

n2 -i

(n

l np

1

=

log I'

n-1

+21og(v-2)

(v+v) -,(1) (n 2 i }

206


and

V +P6 P

_+ XY+p)2

as n -> oo and v -> oo. Furthermore,

(n-1)T(X;R)

-4

X2p(p-1)/2

and

n-16

N

O,tr(B2) -p 2

as n -* oo, where B = Ad 1/2AAd 1/2 with Ad denoting the diagonal matrix of A. In the latter limit, it is assumed that v is known. When v is unknown, the limit still holds for n sufficiently large. The exact distribution of 6 is quite complicated to obtain in a closed form.

10 Estimation

The material in this chapter is of special interest to researchers attempting to model various phenomena based on multivariate t distributions. We shall start with a popular result in the bivariate case.

10.1 Tiku and Kambo's Estimation Procedure In Chapter 4, we studied a bivariate t distribution due to Tiku and Kambo (1992) given by the joint pdf {1+(x2 - µ2)z

1

f (X1, X2)

=

aio2

ko2

k (1 - p2) 1

x exp

2o, (1- p2)

Ql

xi - Al - p2 (x2 - µ2) z (10.1)

Here, we discuss estimation of the parameters p1i 42i U1, a2, and p when v is known. The method for estimating the location and scale parameters

developed by Tiku and Suresh (1992) is used for this problem. For a random sample {(X1i, X2i), i = 1,... , n} from (10.1), the likelihood function is L

oc

{ai o (1

n/2

- p2) }

n

j

{ 1 + X(z:i) - µz 1

1

2Q 1 ( 1

-

-V

k0,2

P2) i=1

{Xii:i - P1 -

}2] z

(X(z:i) - µ2)

where k = 2v - 3, X(2:i), i = 1, ... , n are the order statistics of X2i and X[l:i], i = 1, ... , n are the corresponding concomitant X1 observations. Consider the following three situations: 207

Estimation

208

(i) Complete samples are available and v is not too small (v > 3). (ii) Complete samples are available but v is small (v < 3). (iii) A few smallest or a few largest X2i observations and the corresponding concomitant X[l:i] are censored due to the constraints of an experiment. This situation arises in numerous practical situations. In a time mortality experiment, for example, n mice are inoculated with a uniform culture of human tuberculosis. What

is recorded is X2i: the time to death of the first A(< n) mice, and Xli: the corresponding weights at the time of death. These situations also arise in the context of ranking and selection (David,

1982). We provide some details of the inference for situation (i) as described in Tiku and Kambo (1992). Using a linear approximation of the likelihood based on the expected values of order statistics, it is shown that the maximum likelihood estimators are

Ql

xl--- (x-µ2)i

=

µl

Q2

=

8

1+

= x2 -

µ2

S2 12

2

2 \Qs2

- 1)

s2 2

Pa2 (x1

-'al),

1

Q2

2 +2 I 2

= uu

S1

- 1J

\S1

and

P=

s12 Q2

2 all

where (xl, x2) are the usual sample means, (s2, s2) are the usual sample variances, and s12 is the sample covariance. The estimators µl, µ2, al, 62 i and p are asymptotically unbiased and minimum variance bound estimators. The estimator Qi is always real and positive while the es-

timator p always assumes values between -1 and 1. The asymptotic variances and covariances of the estimators can be written as

V

( \\\

Vl 0

0

I

V2/'

10.1 Tiku and Kambo's Estimation Procedure

209

where

vi

paia2

Pa2ai

a2

1

vl =

n

2mv - nk ( P2aia2 Pal a2 2vmna2 \ Pat orl a2

/

(10.2)

is positive definite and is the asymptotic variance-covariance matrix of (µi, µ2) while pal a2

a2

Pai (l- p2) Pat (1 - P )

v2

Poll (1 - p2)

pa2 (1 - p2)

2 (1 - p2)2

Pa1 s 2

a21

1

=2n

8

2n(2 + 6)

P4ai p2ala2

P2aia2

P3a1 (1 - p2)

pa2 (1 - p2)

a2

p2)

P3a1 (1 p92 (1 - p2)

p2 (1 - p2)2 ) (10.3)

is positive definite and is the asymptotic variance-covariance matrix of (vl, &2i p). The parameters m and 6 are determined by the linear approximation of the likelihood. Interestingly, Var(µl) and Var(µ2) decrease with increasing p2 unless v = oo. The first component on the right of (10.2) is the variance-covariance matrix of µ1, and I2 under bivariate normality, and the first component on the right of (10.3) is the asymptotic variance-covariance matrix of a,, Q2, and p under bivariate normality. The second components in (10.2) and (10.3) represent the effect of nonnormality due to the family (10.1). The asymptotic distri-

bution of J(µl -µl,µ2 - lie) is bivariate normal with zero means and variance-covariance matrix nVl. For testing Ho : (µl,µ2) = (0, 0) versus H1 : (µl,112) 54 (0, 0), a useful statistic is T _ (µl, µ2)T Vl (µl,µ2), the asymptotic null distribution of which is chi-squared with degrees of freedom 2. The asymptotic nonnull distribution is noncentral chisquared with degrees of freedom 2 and noncentrality parameter AV

where A,,.

=

n 1

= a- +

(2kmvn

- 1)

)2' \(92

2

{()2(2(\\f\ (0'2)(L-) 'al a1

Note that A is the noncentrality parameter of the asymptotic nonnull distribution of the Hotelling's T2 statistic based on the sample means

Estimation

210

(ii, x2), sample variances (32, s2), and the sample correlation coefficient 812/(8182). Tiku and Kambo (1992) also provided evidence to the fact that the use of T2 in place of the Hotelling's T2 statistic can result in a substantial gain in power.

10.2 ML Estimation via EM Algorithm Consider fitting a p-variate t distribution to data x1,.. . , x,, with the log-likelihood function

L (µ, R, v)

9 log jRI

-v

p E log (v + si) ,

(10.4)

i=1

where si = (x - µ)TR-1(x - µ) and v is assumed to be fixed. Differentiating (10.4) with respect to µ and R leads to the estimating equations

µ = ave{wixi}/ave{wi}

(10.5)

R = ave{wi(x-µ)(x-µ)T},

(10.6)

and

where wi = (v + p)/(v + si) and "ave" stands for the arithmetic average over i = 1,2,. .. , n. Note that equations (10.5)-(10.6) can be viewed as an adaptively weighted sample mean and sample covariance matrix where the weights depend on the Mahalanobis distance between

xi and µ. The weight function w(s) = (v + p)/(v + s), where s = (x - µ)'R-'(x - µ), is a decreasing function of s, so that the outlying observations are downweighted. Maronna (1976) proved, under general assumptions, the existence, uniqueness, consistency, and asymptotic normality of the solutions of (10.5)-(10.6). For instance, if there exists a > 0 such that, for every hyperplane H, Pr(H) < p/(v + p) - a, then (10.5)-(10.6) has a unique solution. Also, every solution satisfies the consistency property that R) = (µ, R) with probability 1.

The standard approach for solving (10.5)-(10.6) for µ and R is the popular EM algorithm because of its simplicity and stable convergence (Dempster et al., 1977; Wu, 1983). The EM algorithm takes the form of iterative updates of (10.5)-(10.6), using the current estimates of µ and R to generate the weights. The iterations take the form ave {w(m)xi} /ave {w(m) }

10.2 ML Estimation via EM Algorithm

211

and

R(m+1)

= ave { w(m) (xi

- µ(m+1)) (xi

µ(m+1))TI,

-

where w(m) z

_ (v + p)

(v + (x2 - µ(m))T (R(m))

(xi

l

- µ(m))Jf

This is known as the direct EM algorithm and is valid for any v > 0. For details of this algorithm see the pioneering papers of Dempster et al. (1977, 1980), Rubin (1983), and Little and Rubin (1987). Several variants of the above have been proposed in the literature, as summarized in the table below.

Algorithm

Extended EM Restricted EM MC-ECM1 MC-ECM2 ECME1 ECME2 ECME3 ECME4 ECME5 PXEM

Primary References

Kent et al. (1994), Arsian et al. (1995) Arsian et al. (1995) Liu and Rubin (1995) Liu and Rubin (1995), Meng and van Dyk (1997) Liu and Rubin (1995), Liu (1997) Liu and Rubin (1995) Liu and Rubin (1995), Meng and van Dyk (1997) Liu and Rubin (1995), Liu (1997) Liu (1997) Liu et al. (1998)

Consider the maximum likelihood (ML) estimation for a g-component mixture of t distributions given by g

f (x;

E 7rif (xi l-L4, Ri, vi)

,

i=1

where

f (x; tti, Ri, vi)

r'((vi+p)/2) (irvi)p/2 r (vi/2)

IRiI1/2

T X 1+ (x - Ai)Ri (x-N'i) 1

vi

-(v:+p)/2

Estimation

212

R1)T,.. ., (µg, R9))", and v = = (in , ... , 7fg-1, 0T , VT) T, 0 = ((pr, (v1, ... , v9)T . The application of the EM algorithm for this model in a ,p

clustering context has been considered by McLachlan and Peel (1998) and Peel and McLachlan (2000). The iteration updates now take the form n

m

TZj

j=1

m

Uij

In xj

j=1

m Tzj

m

uzj

and R(m+1)

_

E j=1

(7)U gym) (xj _ µ(m+1)) (x7 -µ2m+1)1 T J l

J`

rim), j=1

where u(in) aj

_

lT

1

(xj

(xj

RZm)

v( m) +

- µ2m)/

and

TZj

=

(m') f (xj I-RZm), RZm), v(m))

f (xj; F(m)) The EMMIX program of McLachlan et al. (1999) for the fitting of normal mixture models has an option that implements the above procedure for the fitting of mixtures of t-components. The program automatically

generates a selection of starting values for the fitting if they are not provided by the user. The user only has to provide the data set, the restrictions on the component-covariance matrices (equal, unequal, diagonal), the extent of the selection of the initial groupings to be used to determine the starting values, and the number of components that are to be fitted. The program is available from the software archive StatLib or from Professor Peel's homepage at the Web site address http://www.maths.uq.edu.au/-gjm/

10.3 Missing Data Imputation When a data set contains missing values, multiple imputation for missing data (Rubin, 1987) appears to be an ideal technique. Most importantly,

it allows for valid statistical inferences. In contrast, any single imputation method, such as filling in the missing values with either their

10.3 Missing Data Imputation

213

marginal means or their predicted values from linear regression, typically leads to biased estimates of parameters and thereby often to an invalid inference (Rubin, 1987, pages 11-15).

The multivariate normal distribution has been a popular statistical model in practice for rectangular continuous data sets. To impute the missing values in an incomplete normal data set, Rubin and Schafer (1990) (see also Schafer, 1997, and Liu, 1993) proposed an efficient method, called monotone data augmentation (MDA), and implemented it using the factorized likelihood approach. A more efficient technique to implement the MDA than the factorized likelihood approach is provided by Liu (1993) using Bartlett's decomposition, which is the extension of the Bayesian version of Bartlett's decomposition of the Wishart distribution with complete rectangular normal data to the case with monotone ignorable missing data.

When a rectangular continuous data set appears to have.longer tails than the normal distribution, or it contains some values that are influential for statistical inferences with the normal distribution, the multivariate t distribution becomes useful for multiple imputation as an alternative to the multivariate normal distribution. First, when the data have

longer tails than the normal distribution, the multiply imputed data sets using the t distribution allow more valid statistical inferences than those using the normal distribution with some "influential" observations deleted. Second, it is well known that the t distribution is widely used in applied statistics for robust statistical inferences. Therefore, when an incomplete data set contains some influential values or outliers, the t distribution allows for a robust multiple imputation method. Furthermore, the multiple imputation appears to be more useful than the asymptotic method of inference since the likelihood functions of the parameters of the t distribution given the observed data can have multiple modes. For a complete description of the MDA using the multivariate t distribution, see Liu (1995). See also Liu (1996) for extensions in two aspects, including covariates in the multivariate t models (as in Liu and Rubin, 1995), and replacing the multivariate t distribution with a more general class of distributions, that is, the class of normal/independent distributions (as in Lange and Sinsheimer, 1993). These extensions provide a flexible class of models for robust multivariate linear regression and multiple imputation. Liu (1996) described methods to implement the MDA for these models with fully observed predictor variables and possible missing values from outcome variables.

Estimation

214

10.4 Laplacian T-Approximation The Laplacian T-approximation (Sun et al., 1996) is a useful tool for Bayesian inferences for variance component models. Let p(0 I y) be the posterior pdf of 0 = (01i . . . , BP)T given data y, and let 77 = g(0) be the parameter of interest. Leonard et al. (1994) introduced a Laplacian T-approximation for the marginal posterior of ij of the form

x IT,I-112 p Y) A w/2 f (i 1w, 0n, T,) to be the marginal posterior pdf of q, where (e,11

p* (,qI Y)

11

T,n

=

U71

(w + p) Aq

TI

77

11

w+p-1'

U,? +w+p-1'

Qn

1 71

W

_ 1TQ-11

All

(10.7)

=

=

0**

alogp(01 Y) a0

0_0,

a2logp(0 I Y) a (99T)

=

0=0,,

0,1 +

w, 0*, T.n) denotes the pdf of q = g(0) when 0 possesses a multivariate t distribution with mean vector 0*, covariance matrix T,,, and degrees of freedom w. Here, 0,, represents some convenient and f (,q

I

approximation to the conditional posterior mean vector of 0, given q, and w should be taken to roughly approximate the degrees of freedom of a generalized multivariate T-approximation to the conditional distribution of 0 given q. When 0,, is the conditional posterior mode vector of 0, given rj, (10.7) reduces to the Laplacian approximation introduced by Leonard (1982)

and shown by Tierney and Kadane (1986) and Leonard et al. (1989) to possess saddlepoint accuracy as well as an excellent finite-sample accuracy, in many special cases. It was previously used for hierarchical models by Kass and Steffey (1989).

10.5 Sutradhar's Score Test

215

In the special case where q = aT B is a linear combination of the 0's, the approximation (10.7) is equivalent to p* (rl IY)

a

IT,

1-112

p

(0,71

Y)

),-(,, +p) /2tn (w,

aTo*, (aTT,a) -1)

where t, (w, p, T) denotes a generalized t pdf.

10.5 Sutradhar's Score Test Consider a random sample X1,. .. , Xn from a p-variate t distribution with the pdf

f (x) = 7

(v - 2)"/2IF ((v + p)/2) IRI1/2

7rp/2I' (v/2)

x

[v-2+(xj-µ)TR-1(xj-p)]

.

Note this is a slight reparameterization of the usual t pdf. The loglikelihood n

G = Elog f (xj) j=1

is a function of the parameters R, p, and v. Frequently in social sciences, and particularly in factor analysis, one of the main inference problems is to test the null hypothesis R = Ro when µ and v are known. Sutradhar (1993) proposed Neymann's (1959) score test for this test for large n. Le r = (rl1, ... , rhl, ... , rpp)T be the p(p+ 1)/2 x 1 vector formed by stacking the distinct elements of R, with rhi being the (h, l)th element of the p x p matrix R. Also let Ap(p+l)/2)

(A,,...,A

T

=

b (roe A, v)

and

(ro,N-,-0) 1

T

ro>

>v

where b(ro, µ, v), (ro, v), and q(ro, µ, are the score functions obtained under the null hypothesis r = ro, by replacing µ and v with their consistent estimates µ and v in b (ro

v)

=

ar '

(10.8)

Estimation

216

(ro

e

µ v)

=

aG aµ '

(10 . 9)

and

aG

(10 . 10)

av,

respectively. Furthermore, let Ti(ro, µ, v) = Ai - EP±i /3ij ryj, where /3i,j is the partial regression coefficient of Ai on y3. Then, Neyman's partial score test statistic is given by W (µ, v)

TT Mil

- (212M13)

MT12 M3 3

T

T,

M13 (10.11)

where T =_ [Ti (ro,

Tp(p+l)/2(ro ) µ, v)]T for i, r = 1, 2, 3; Mir

are obtained from Mir = E(-Dir) by replacing it and v with their consistent estimates; and Dir for i, r = 1,2,3 are the derivatives given by a2G D11

D12

D13 =

arar" a2G

araµ" a2G arav'

a2G

D22 = aµaft" a2G D23

aµav'

and

D33 =

02G av2

Under the null hypothesis r = ro, the test statistic W (A, v) has an approximate chi-squared distribution with degrees of freedom p(p + 1)/2. The test based on (10.11) is asymptotically locally most powerful. Clearly the implementation of this test requires consistent estimates of

10.5 Sutradhar's Score Test

217

µ, v as well as expressions for the score functions and the information matrix. The maximum likelihood estimates of µ and v are obtained by simultaneously solving n

tgj1Xj j=1

Eq.,-.l

j=1

and 71 (A, ro, = v) 0,

where qj = v - 2 + (Xj - µ)TRo(Xj - µ) and Ro is specified by the null hypothesis. The moment estimates of µ and v (which also turn out to be consistent) are n 1

Xj j=1

and

2 {2/32 - f (ro)J

02 - f (ro) where n

n(Xj-X)TRo(Xj-X)]

l2

j=1

is a consistent estimator of the multivariate measure of skewness (see, for example, Mardia, 1970b), and f (ro)

_ 3 E (roh)2 {rhh } 2 + h =1

12

{rhh hu h'

1

rohro h + (roh )2I ,

l

where rhh, and ro " denote the (h, h')th element of Ro and Ro 1, respectively.

10.5.1 Score Functions The score functions defined in (10.8), (10.9), and (10.10) are given by

b(r,Fi,V)

_ -2 InIp- (v+p)R-1Eq,-.1Aj I R-1, j=1

Estimation

218

n p)R-' E qj_1

(v +

(Xj - N-)

,

j=1

and

n{22 +log(v_2)+b()

ij (r, µ, v)

2b\2I]

-12 I(V+P) +Eloggj j=1

q,,

j=1

respectively, where denotes the digamma function and qj is a nonhomogeneous quadratic form given by qj = v - 2 + trR-1 B j with B j =

(X3 - tt)(Xj - µ)'.

10.5.2 Information Matrix By taking the second derivatives and then applying expectations, one can derive the elements of the information matrix. The first element takes the complicated form [m* (1,1), m* (1, 2), ... , m* (h, l), ... , m* (p, p)]

M11 =

,

where, for 1 > h, h, 1 = 1,. .. , p, m*(h, l) is the p(p + 1)/2-dimensional vector, formed by stacking the distinct elements of the p x p symmetric matrix n Mn,1

=

2

[rh

T

®(rl) J -

nv+ (2 p

R-1Qn,cR-1.

Here, r'k denotes the kth column of the R-1 matrix, and the (u, v)th element of the p x p matrix Qh,t is given by (v + 2) 2

P

P

+r r

VVrhZrtk (r r UV (v+4)2(v+p)(v+p+2)

Zu

v

+r

Zv

n- U )

'

i=1 j=1

where rms and rn,,s denote the (m, s)th element of R-1 and R, respectively. The second element of the information matrix M12 is zero. The third element M13 is formed by stacking the distinct elements of the symmetric matrix

n(p+2) (v - 2)(v + p)(v + p + 2)

R

1 .

10.6 Multivariate t Model

219

The remaining elements of the information matrix are given by

nv(v + p) (v - 2) (v +p+2) R

M22

M23 =

1

'

0,

and 1

M33

[i,(v+p\

4

, (l \21

v-4 2(v - 2)2

lnv(v2+vp-6p-2v-8)

+2 (v-2)2(v+p)(v+p+2) 10.6 Multivariate t Model Consider the following multivariate t model described in equation (9.2) of the preceding chapter

f (xi,...,X.)

-

F ((v + p)/2)

(irnv)n12 r (v/2)

IRIn12

n

x 1+ 1 V

t (xi - µ)T R-1 (xi - µ)

-(v+nv)12 .

(10.12)

i=1

In this section, we consider estimation issues associated with the corre-

lation matrix R and its trace tr(R).

10.6.1 Estimation of R Joarder and Ali (1997) developed estimators of R (when the mean vector µ is unknown) under the entropy loss function

L (u (A), R) = tr (R-1u (A)) - log IR-1u (A) I - p, where u(A) is any estimator of R based on the Wishart matrix A defined in (9.4). Based on the form of the likelihood function, the entropy loss function has been suggested in the literature by James and Stein (1961)

and is sometimes known as the Stein loss function. Some important features of the entropy loss function are that it is zero if the estimator u(A) equals the parameter It, positive when u(A) R, and invariant under translation as well as under a natural group of transformations of covariance matrices. Moreover, the loss function approaches infinity as

Estimation

220

the estimator approaches a singular matrix or when one or more elements (or one or more latent roots) of the estimator approaches infinity. This

means that gross underestimation is penalized just as heavily as gross overestimation. In estimating R by u(A), Joarder and Ali (1997) considered the risk function R(u(A), R) = E[L(u(A), R)]. An estimator u2(A) of R will be said to dominate another estimator ul (A) of R if, for all R belonging to the class of positive definite matrices, the inequality R(u2 (A), R) < R(ul (A), R) holds and the inequality R(u2 (A), R) < R(ul (A), R) holds for at least one R. Joarder and Ali (1997) obtained three estimators for It, by minimizing the risk function of the entropy loss function among three classes of estimators.

First, it is shown that the unbiased estimator ft = (v - 2)A/(vn) has the smallest risk among the class of estimators of the form cA, where c > 0, and the corresponding minimum risk is given by

R (R, R) = p log n - E E [log (xn+i_i) ] + p log 2

=1

\

-2pE (logy) , where r has the inverted gamma distribution given by (9.3). Second, the estimator R* = TD*TT, where T is a lower triangular matrix such that A = TTT and D* = diag(di,... , d p*) with dz defined by d*

_ -

v-2

Z

1

v n+p+1-2i'

has the smallest risk among the class of estimators TATT, where A belongs to the class of all positive definite diagonal matrices and the corresponding minimum risk function of R* is given by P

R (R*, R) =

P

log(n + 1 +p - 2i) +p log

v 2

E [log

(xn+1_

)]

2pE (log rr) ,

where T is as defined above. Furthermore, R* dominates the unbiased

estimator R = (v - 2)A/(vn).

10.6 Multivariate t Model

221

e Finally, consider the estimator R = SO(M)S, where A has the spectral decomposition A = SMST, with ¢(M) = D*M. Then the estimator R = SD* MST dominates the estimator R* = TD*TT.

10.6.2 Estimation of tr(R) Let 6 = tr(R) denote the trace of R. Joarder (1995) considered the estimation of 6 for the multivariate t model under a squared error loss function following Dey (1988). The usual estimator of 6 is given by 8 = cotr(A), where co is a known positive constant and A is the Wishart matrix defined in (9.4). The estimator 8 defines an unbiased estimator of 8 for co = (v - 2)/(vn) and a maximum likelihood estimator of 8 for co = 1/(n + 1) (see, for example, Fang and Anderson, 1990, page 208). Joarder and Singh (1997) proposed an improved estimator of 6 - based on a power transformation - given by

=

8

cotr (A) + coc {p

IAI1/r - tr (A) }

,

where co is a known positive constant and c is a constant chosen so that the mean square error (MSE) of 8 is minimized. Calculations show that

MSE (8)

= MSE (8)

+ c/31 + c2/92,

where 01

=

2co2E [(cotr (A) - 8) (p lAl1/n

- tr (A))]

(10.13)

and

02

= coE

[p jAj1/P

- tr (A)]

.

(10.14)

Thus MSE(8) is minimized at c = -/31/(2/92) and the minimum value is given by MSE(8) - /31 /(4/32). This proves that 6 is always better than the usual estimator in the sense of having a smaller MSE. The estimate of c is given by c = -/31/(2/92), where /91 and /32 are obtained by calculating the expectations in (10.13) and (10.14) using the numerous properties given in Section 9.1 and then replacing R by the usual estimator c0A.

It can be noted from Fang and Anderson (1990, page 208) that the estimators /91 and /32 are the maximum likelihood estimators of /31 and /32i respectively, provided R = c0A and co = 1/(n + 1). The following table taken from Joarder and Singh (1997) presents the percent relative efficiency of 8 over 8.

Estimation

222

R = diag(1,1,1)

R = diag(4, 2,1)

R = diag(25,1,1)

105.32 102.13

130.31

153.90

10

117.56

148.76

15

101.53

112.07

127.15

V

5

The numbers are from a Monte Carlo study carried out by generating 100 Wishart matrices from the multivariate t-model with n = 25 and p = 3.

10.7 Generalized Multivariate t Model Consider the generalized multivariate t model (9.7) discussed in the pre-

ceding chapter. The usual estimator of R is a multiple of the Wishart matrix of the form R = coA, where co > 0. Joarder and Ahmed (1998) proposed improved estimates for R as well as its trace and inverse under the quadratic loss function. The proposed estimators for R are

R = coA - c IAI1/P I,

(10.15)

where I is an identity matrix and c is chosen such that R is positive definite. For an estimator R* of R, let L(R*, R) = tr[(R* - R)2] denote the quadratic loss function and let R(R*, R) = EL(R*, R) denote the corresponding risk function. The relationship between R and R is rather involved. Defining the dominance of one estimator over another in the same manner as in Section 10.6.1, Joarder and Ahmed (1998) established that R dominates R for any c satisfying d < c < 0, where conp+2 p

d

FP((n-1)/2+1/p)

(10.16)

y FP ((n - 1)/2 + 2/p)

with co < pry/((n - 1)p + 2) or 0 < c < d, where d is given by (10.16) with co > pry/(np+ 2) and y by y = 72 /74 and y2 = E(rri), i = 1, 2, 3, 4. The risk functions of the two estimators are given by

R (R-, R) =

4p'Y4

- d Ir (R/P)

FP (Pl( 2/p) IRI2/P C 1\

1/P

Cc

+ {1 + (n - 1)coy4 (con - 2y)} tr (R2) +(n - 1)c02y4 (trR)2 and

R (R, R) =

{1 + (n - 1)coy4 (con - 2y)} tr (R2)

10.8 Simulation +(n - 1)c02ry4 (trR)2

223 ,

respectively. Now consider estimating the trace S = trR. The usual and

the proposed estimators are S = cotrA and 8 = cotrA - cp I A Il/p respectively, where co > 0 and c is such that the proposed estimator is positive. Joarder and Ahmed (1998) established that the corresponding risk functions are given by

R(8,6) = [(n-1)co{(n-1)coy4-2'y2}+1]82 +2(n - 1)coy4tr (R2) and

R (S, 8)

+ 2/p) IRI2iP = R (d S) + 4p2y4 Fp rp(n/2(n/2)

C

c - tr (R/p) d IRI1/P

respectively. It is evident that S dominates 8. Finally, consider estimating the inverse %F = R-1 with the usual and the proposed estimators = coA-1 and 'F = given by - co I A I-1/p I, respectively, where co > 0 and c is such that the _proposed estimator is positive definite. In this case, it turns out that 'F dominates ;P- for any c satisfying coA-1

d no, this expression reduces to the power of the F test under normality (Tiku, 1967). The distribution of future responses given a set of data from an informative experiment is known as a predictive distribution. Haq and Khan (1990) derived the predictive distribution for (11.1). Rewrite (11.1) in the equivalent form y = 8X + ae and let Yf be a future response corresponding to the design vector x f, that is, yf = /3xf + ae f. Haq and Khan (1990) showed that the predictive pdf of Yf is given by f (Yf oc

I Y)

(n-p+1)/2

[i + s-2(Y) {yf - b(Y)xf} (1 - xfTA-'xf) {yf - b(Y)xf IT J

where b(e) = eXT(XXT)-1, s2(e) = (e - ;&) (e - e) T, e = b(e)X, and A = XXT + x fxf . Thus, for the given informative data y, the predictive distribution of Yf is t with mean vector b(y)x f, variance-covariance matrix (n - p)s2(y)/{(n - p - 2)(1 - XT A-1x f)}, and degrees of freef dom n - p. A prediction interval of the desired coverage probability can easily be obtained by using the standard t-table. Note that the predictive distribution does not depend on the degrees of freedom parameter of the original t distribution. For a set of n' future responses given by

11.2 Bayesian Linear Models

233

Yf = )3X f + ore f, Haq and Khan (1990) noted similarly that the predictive distribution of y f is n'-variate t with mean vector b(y)Xf, variancecovariance matrix I I,,,, - XfQ-1X f-1/2 s(y) (where Q = XXT + Xf X f ), and degrees of freedom n - p. It is to be noted that the distri-

bution of (n-p)s-2(y)(Yf -b(Y)Xf)(In,' -XfQ-1Xf)(Yf -b(Y)Xf)T

is F with degrees of freedom n', and n - p. This distribution can be utilized for determining the prediction region for a set of future responses with any desired coverage probability.

11.2 Bayesian Linear Models In his classical paper, Zellner (1976) provided a Bayesian analysis of the linear model (11.1). Consider the diffusion prior for /3 and Q2, that is,

p(/i,a2)

oc

(11.11)

T2 ,

where 0 < o, 2 < oo and /3a E R, i = I,-, k. Then, assuming that v is known, the posterior pdf of the parameters is

a { A (N) } -n/2

P (Q, Q 2 I Y, V)

[vat /A (Q)] A (i3) [1 + vat /A

v/2-1 (,8)](,,+, )/2 1

where A(,Q) = (y - X)9)T (y - X,3). It follows that the conditional posterior pdf of ,Q given a2 and v is in the form of a multivariate t pdf with mean /j (the least squares estimate in (11.2)). The corresponding conditional posterior covariance matrix is given by vo2 + (n - p)s2

T

v+n - P - 2 (X X)

-1

provided that n - p + v > 2, where (n -p)82 = (y - X/3)T (y - X/l). As v -+ oo, the conditional posterior pdf for ,Q and a2 approaches a mul(XTX)-1a2, tivariate normal pdf with mean ,Q and covariance matrix which is the usual result for the normal regression model with the diffuse prior pdf (11.11). The marginal posterior pdf for ,Q is _ yn'2, ( {(n-p)s2+(/i-,Q)7

P(/jJY,v)

oc

XTX (13 -)

(11.12)

which is in the form of a p-dimensional t pdf and does not depend on the value of v. In fact, (11.12) is precisely the result that one obtains in the Bayesian analysis of the normal regression model with the diffuse

Regression Models

234

prior for the parameters shown in (11.11). The marginal posterior pdf for a2 is

all (v/2)-1 p

(0,21

y, v)

\ s2 /

-(v+n-p)/2

vO,2

(\1 +

OC

(n - p)s2 )

from which it follows that a2/s2 has the F pdf with degrees of freedom v and n - p, a result paralleling the classical results mentioned in the preceding section. From properties of the F distribution, the modal

value of o, 2/s2 is ((n - p)/v)((v - 2)/(n-p+2)), when v > 2 and its mean is (n - p) / (n - p - 2) when n - p > 2. Also, as v -4 oo, the posterior distribution of vs2/v2 approaches a chi-squared distribution with degrees of freedom n - p, a distributional result that holds for the Bayesian analysis of the usual normal regression model with diffuse prior assumptions. Finally, note that the posterior pdf for no'2 / (y - XA)T (y X,3) is F,,,n. The natural conjugate prior distribution for o, 2 and ,8 is the product of the marginal F pdf for a2 times a conditional p-dimensional t pdf for ,3 given Q2, that is, P (Q, Or2I ')

= PF (0,21 _) Ps (QI a2, ')

,

where 2

PF

(01 2

2

2) (v-2)/2

(Vol /vasa)

say vas v)

oc vQ2/vaSa)(v+va)12

(1 +

(where va > 0, Sa > 0, and 0 < a < oo) and

PS (QI

2

Q, A, I/a)

oc

Oa P f a +

1

Q2 a

(-

(2v+P)l2

)T

A (Q

where /32 E R, i = 1, ... , p, A is symmetric and positive definite, fi is

the prior mean vector, 1/a = v + va, and va = (vasa + vat)/Va. As with the natural conjugate for the usual normal regression model, it is seen that A and u2 are not independent in the natural conjugate prior distribution in (11.13). If the natural conjugate prior distribution is thought to represent the available prior information adequately, it can be used for obtaining the posterior distribution; see the appendix in Zellner (1976) for details.

11.3 Indexed Linear Models

235

11.3 Indexed Linear Models Lange et al. (1989) and Fernandez and Steel (1999) provided a farreaching extension of (11.1) to handle the situation when yj's are assumed to have the t distribution with degrees of freedom vi and parameters 1z = gi (O) and R = hi (p) indexed by some unknown parameters 0 and 0. Lange et al. (1989) suggested an EM algorithm for estimation. They also considered methods for computing standard errors, developed graphical diagnostic checks, and provided applications to a variety of problems. The problems include linear and nonlinear regression, robust estimation of the mean and covariance matrix with missing data, unbalanced multivariate repeated-measures data, multivariate modeling of pedigree data, and multivariate nonlinear regression. They also derived the expected information matrix for (0, 0, v) for one observation in the form

_

E 8 log L 1

V+ P avT

v+p+2 aoi

C99 99

a log L

l aoiaoj/

_

V+ P

2(v+p+2)

tr R-1

(

2 1 2 (v+p+)

E

alogL)

__

tr

81ogL

1

1

2

v+p

2( 2) 1

1

R-1

1' (vl 2

OR1

a0i p

\2/ + v(v+p)

v+2

v+p+v(v+p+2)

where

a0,'

R-1 aR tr R-1 s:),

(v+p)(v+p+2)tr

_

av

9R R_1 aR a0; a0i

00i0v

E (avav)

R-1

'

0 (x) = d2 log F(x)/d2x is the trigamma function. The remaining

elements of the matrix are zero.

In an important paper, Fernandez and Steel (1999) revealed some pitfalls with a model of the above kind. Under a commonly used noninformative prior, they showed that Bayesian inference is precluded for certain samples, even though there exists a well-defined conditional distribution of the parameters given the observables. They also noted that global maximization of the likelihood function is a vacuous exercise since

236

Regression Models

the latter becomes unbounded as one tends to the boundary of the parameter space. More specifically, let 1(9, a, R, v) be the likelihood function for n independent observations yj assumed to have the t distribution with mean vector gi(9), common covariance matrix Q2R, and common degrees of freedom v. For given values of 0 = Bo, R = Ro, and v = vo, let 0 < s(9o) < n denote the number of observations for which yi = gi(9o). Then the following hold

(a) If vo

0 and p(v) is proper and functionally independent of (T, /3, rj), if the posterior of v exists, then p(v I y, X) = p(v).

12 Applications

Due to limitations on the size of this book and since the aim is to collect and organize results on multivariate t distributions, in this short chapter we collect and present a small number of relatively recent applications

of multivariate t distributions. The treatment is by no means exhaustive. Some other applications - in particular those related to Bayesian inference - are mentioned in the previous chapters (see Chapters 1, 3, 5, 10, and 11).

12.1 Projection Pursuit Exploratory projection pursuit is a technique for finding "interesting" low p-dimensional projections of high P-dimensional multivariate data; see Jones and Sibson (1987) for an introduction. Typically, projection pursuit uses a projection index, a functional computed on a projected density (or data set), to measure the "interestingness" of the current projection and then uses a numerical optimizer to move the projection direction to a more interesting position. Loosely speaking, a robust projection index is one that prefers projections involving true clusters over those consisting of a cluster and an outlier. A good robust projection index should perform well even when specific assumptions required for "normal operation" fail to hold or hold only approximately. In a paper that was awarded the Royal Statistical Society Bronze Medal, Nason (2001) developed five new indices based on measuring divergence from the multivariate t distribution with the joint pdf f (x)

-

I' ((v +p)/2) 7fp/2 (v - 2)p/2r' (v/2) (1 241

xTx

v - 2)

- (-+p)/2

Applications

242

that are intended to be especially robust. The first three indices are all weighted versions of the L2-divergences from f for v > 3. They are given by

ITa

2

f{g(x)_f(x)}2f(x)dx

=

for a = 0, 1/2, 1. Nason (2000) derived an explicit formula for the case a = 0. The fourth index is the Student's t index defined by

ITI

fgl_2/(V+P)(x)dx.

V

This index is minimized over all spherical densities by f (x). Specifically, it satisfies the inequality

(v+p-2)F((v+p)/2)

>

-

IV

7rp/2(v - 2) 1+p/211 (v/2)

for all spherical densities g with equality if and only if g = f almost everywhere. The proof of this result uses the fact that the index can be represented as the sum of two F-divergences (Vajda, 1989). Through both numerical calculation and explicit analytical formulas, Nason (2001)

found the the Student's t indices are generally more robust and that indices based on L2-divergences are also the most robust in their class. A detailed analytical exploration of one of the indices (ITo 2) showed that it acts robustly when outliers diverge from a main cluster but behaves like a standard projection index when two clusters diverge, that is, its behavior automatically changes depending on the degree of outlier contamination. The degree of sensitivity to outliers can be reduced by increasing the degrees of freedom v of the ITo2 index to make it behave increasingly like Hall's index (Hall, 1989) as v -4 oo.

Using the transformation x = tan(g), Nason further developed the orthogonal expansion index given by 2

ITL2

=

V

x

/2

2

f ir/2 l go (0) - x{

2

cos4 9 } dB, JJJ

where go is the pdf of the transformed projected data X. Using the Fourier series expansion of go(9) on [-it/2,7r/2], go (0)

=2+

{ancos(2n0) + an sin(2n8)} n=1

,

12.2 Portfolio Optimization

243

where

an =

2

it/2

g(O) cos(2n9)dO -7r/2

and 2

lr/2

IT

-7r/2

=

bn

g(O) sin(2n9)dO,

the index ITL2 can be expanded as TL2

13,1/2

_

7 2

-

2

3 12 (ao

2

c

-

27

I

+ (al

7r) + (a2

2

47r

)

00

+Ea2n+:L bn n=3

n=3

12.2 Portfolio Optimization There are a number of places in finance where robust estimation has been used. For example, when a stock's returns are regressed on the market returns, the slope coefficient, called beta, is a measure of the relative riskiness of the stock in comparison to the market. Quite often, this regression will be performed using robust procedures. However, there appear to be fewer applications of robust estimation in the area of portfolio optimization. In the problem of finding a risk-minimizing portfolio subject to linear constraints, the classical approach assumes normality without exceptions. Lauprete et al. (2002) addressed the

problem when the return data are generated by a multivariate distribution that is elliptically symmetric but not necessarily normal. They showed that when the returns have marginal heavy tails and multivariate tail-dependence, portfolios will also have heavy tails, and the classical procedures will be susceptible to outliers. They showed theoretically, and on simulated data, that robust alternatives have lower risks. In particular, they showed that when returns have a multivariate t distribution with degrees of freedom less than 6, the least absolute deviation (LAD)

estimator has an asymptotically lower risk than the one based on the classical approach. The proposed methodology is applicable when heavy tails and tail-dependence in financial markets are documented especially at high sampling frequencies.

244

Applications

12.3 Discriminant and Cluster Analysis In the past, there have been many attempts to modify existing methods of discriminant and cluster analyses to provide robust procedures. Some of these have been of a rather ad hoc nature. Recently the multivariate t distribution has been employed for robust estimation. Suppose, for simplicity, that one utilizes two samples in order to assign a new observation into one of two groups, and consider the joint distribution

f xl,x2) vl-v --2r (v + np/2) ,np/2

IRIn/2

2

- (v+np)/2

n;

x 1(v - 2) +

(xij i_1 j=1

- µi)TR 1(x2 -µ2)

(12.1)

of the two samples Xi = (X11, ... , X1n1) and X2 = (X21, ... , X2n2) of sizes nl and n2, respectively. In (12.1), n = ni + n2. The (nl + n2)pdimensional t distribution (12.1) was proposed by Sutradhar (1990). It is evident that the marginals are distributed according to f (x2a)

-

v _-217 (v + p/2) 7rp/2 IRIn/2

(v+p)/2

x [(v -

2) + (xij - µi)T R-1 (xij - µi),

(12.2)

which is a slight reparameterization of the usual multivariate t pdf. Let 7r1 and 7r2 denote the two t-populations of the form (12.2) with parameters (µl, R, v) and (µ2, R, v), respectively. Fisher's optimal discrimination criterion is robust against departure from normality (Sutradhar, 1990), and it assigns the new observation with measurement X to 7r1 if d(x)

1

_ (Al - µ2)T R-lx - 1 (Al - µ2)T R-1 (µl + µ2) ? 0;

otherwise, it assigns the observation to 7r2. But even though the classification is based on the robust criterion, the probability of misclassification depends on the degrees of freedom of the t distribution. If el and e2 are probabilities of misclassification of an individual observation from 7r1 into 72 and from 72 into 7rl, respectively, then ei

=

v --2 r (v + 1/2)

-°/2 00

{(v - 2) +

z2}-(v+1)/2

dz

12.4 Multiple Decision Problems

245

for i = 1, 2, where A2 = (Al - µ2)T R-1 (µl - µ2). Calculations of el and e2 for selected values of 0 and v (Sutradhar, 1990) suggest that if a sample actually comes from a t-population (12.2) with degrees of freedom v, then the evaluation of the classification error rates by normal-based probabilities would unnecessarily make an experimenter more suspicious. Sutradhar (1990) illustrated the use of the preceding discrimination approach by fitting the t distribution to some bivariate data on two species of flea beetles. McLachlan and Peel (1998), McLachlan et al. (1999), and Peel and McLachlan (2000) used a mixture model of t distributions for a robust method of mixture estimation of clustering. They illustrated its usefulness by a cluster analysis of a simulated data set with added background noise and of an actual data set. For other recent methods for making cluster algorithms robust, see Smith et al. (1993), Dave and Krishnapuram (1995), Jolion et al. (1995), Frigui and Krishnapuram (1996), Kharin (1996), Rousseeuw et al. (1996), and Zhuang et al. (1996).

12.4 Multiple Decision Problems The multivariate t distribution arises quite naturally in multiple decision problems. In fact, it is one of the earliest applications of this distribution in statistical inference. Suppose there are q dependent variates with means 01 i ..., 9h, ..., 9q, respectively, and that one has estimators 9h of Oh, h = 1, . . . , q available, which are jointly distributed according to a q-variate normal distribution with mean 0h, h = 1, ... , q, and covariance matrix o-2R, where R is a q x q positive definite matrix and 0.2 is an unknown scale parameter. Let s2 be an unbiased estimator of Q2 such that s2 is independent of the 0h's and vs2/Q2 has the chi-squared distribution with degrees of freedom v. Consider p < q linearly independent linear combinations of Oh's, q

mi =

CihOh = cT B, h=1

for i = 1,.. , p, where ci =

(Ci1, ... , Cih, ... , Ciq)T is a q x 1 vector of known constants. The unbiased estimators of the mi's are .

q

mi =

Ciheh = CT B, h=1

Applications

246

each of which is a normally distributed random variable with mean mi and variance cTRci. Then Yi

=

mi - mi s ciT Rci

i = 1,...,p

is a Student's t-variate and Y1, ... , Y, have the usual p-variate t distribution with degrees of freedom v, zero means, and the correlation matrix {8iv} given by cTRc,,,

cTRcic,T.Rcu

For multiple comparisons, one computes the one- and two-sided confidence interval estimates of mi (i = 1, . . . , p) simultaneously with a joint confidence coefficient 1 - a, say. These estimates are given by (Dunnett, 1955)

ii ± h1s cTRci and

i12i ± has

cTRci,

respectively, where the constants h1 and h2 are determined so that the intervals in each case have a joint coverage probability of 1 - a. The constants h1 and h2 can be computed using the methods discussed in Chapter 8.

12.5 Other Applications Bayesian prediction approaches using the multivariate t distribution have attracted wide-ranging applications in the last several decades, and many sources are available in periodic and monographic literature. Chien (2002) discusses applications in speech recognition and online environmental learning. In experiments of hands-free car speech recogni-

tion of connected Chinese digits, it was shown that the proposed approach is significantly better than conventional approaches. Blattberg and Gonedes (1974) were one of the first to discuss applications to security returns data. For other applications, we refer the reader to the numerous modern books on multivariate analysis and to the Proceedings of the Valencia International Meetings.

References

Abdel-Hameed, H. and Sampson, A. R. (1978). Positive dependence of the bivariate and trivariate absolute normal, t, x2, and F distributions, Annals of Statistics 6, 1360-1368. Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions (Dover, New York). Abusev, R. A. and Kolegova, N. V. (2001). On estimation of probabilities of linear inequalities for multivariate t distributions, Journal of Mathematical Sciences 103, 542-546. Aczel, J. (1966). Lectures on Functional Equations and Their Applications (Academic Press, New York). Afonja, B. (1972). The moments of the maximum of correlated normal and t variates, Journal of the Royal Statistical Society B 34, 251-262. Ahmed, A. and Gokhale, D. (1989). Entropy expressions and their estimators for multivariate distributions, IEEE Transactions on Information Theory 35, 688-692. Ahner, C. and Passing, H. (1983). Berechnung der multivaiaten t-verteilung and simultane vergleiche gegen eine kontrolle bei ungleichen gruppenbesetzungen, EDV in Medizin and Biologic 14, 113-120. Amos, D. E. (1978). Evaluation of some cumulative distribution functions by numerical evaluation, SIAM Review 20, 778-800. Amos, D. E. and Bulgren, W. G. (1969). On the computation of a bivariate t distribution, Mathematics and Computation 23, 319-333. Anderson, D. N. and Arnold, B. C. (1991). Centered distributions with Cauchy conditionals, Communications in Statistics-Theory and Methods 20, 2881-2889. Anderson, T. W. (1984). An Introduction to Multivariate Analysis, second edition (John Wiley and Sons, New York). Anderson, T. W. and Fang, K. T. (1987). Cochran's theorem for elliptically contoured distribution, Sankhyd A 49, 305-315. Ando, A. and Kaufman, G. W. (1965). Bayesian analysis of the independent multi-normal process - neither mean nor precision known, Journal of the American Statistical Association 60, 347-358. Arellano-Valle, R. and Bolfarine, H. (1995). On some characterization of the t-distribution, Statistics and Probability Letters 25, 79-85. Arellano-Valle, R., Bolfarine, H. and Iglesias, P. L. (1994). A predictivistic interpretation of the multivariate t distribution, Test 3, 221-236. 247

248

References

Armitage, J. V. and Krishnaiah, R. R. (1965). Tables of percentage points of multivariate t distribution (abstract), Annals of Mathematical Statistics 36, 726.

Arnold, B. C. and Beaver, R. J. (2000). Hidden truncation models, Sankhya A 62, 23-35. Arnold, B. C. and Press, S. J. (1989). Compatible conditional distributions, Journal of the American Statistical Association 84, 152-156. Aroian, L., Taneja, V. and Cornwall, L. (1978). Mathematical forms of the distribution of the product of two two normal normal variates, Communications in Statistics-Theory and Methods 7, 165-172. Arsian, 0., Constable, P. D. L. and Kent, J. T. (1995). Convergence behavior of the EM algorithm for the multivariate t distribution, Communications in Statistics-Theory and Methods 24, 2981-3000. Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution, Journal of the Royal Statistical Society B 61, 579-602. Azzalini, A. and Capitanio, A. (2002). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. Submitted to Journal of the Royal Statistical Society B. Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew normal distribution, Biometrika 83, 715-726. Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions (John Wiley and Sons, Chichester). Bechhofer, R. E. and Dunnett, C. W. (1988). Tables of percentage points of multivariate t distributions, in Selected Tables in Mathematical Statistics 11, ed. R. E. Odeh and J. M. Davenport (American Mathematical Society, Providence, Rhode Island). Bechhofer, R. E., Dunnett, C. W. and Sobel, M. (1954). A two-sample multiple-decision procedure for ranking means of normal populations with a common unknown variance, Biometrika 41, 170-176. Bennett, B. M. (1961). On a certain multivariate normal distribution, Proceedings of the Cambridge Philosophical Society 57, 434-436. Bickel, P. J. and Lehmann, E. L. (1975). Descriptive statistics for nonparametric models I. Introduction, Annals of Statistics 3, 1038-1044. Birnbaum, Z. (1948). On random variables with comparable peakedness, Annals of Mathematical Statistics 19, 76-81. Blattberg, R. C. and Gonedes, N. J. (1974). A comparison of the stable and Student distributions as statistical models for stock prices, Journal of Business 47, 224-280. Bohrer, R. (1973). A multivariate t probability integral, Biometrika 60, 647-654.

Bohrer, R. and Francis, G. K. (1972). Sharp one-sid d confidence bounds over positive regions, Annals of Mathematical Statistics 43, 1541-1548. Bohrer, R., Schervish, M. and Sheft, J. (1982). Algorithm AS 184: Non-central studentized maximum and related multiple-t probabilities, Applied Statistics 31, 309-317. Bowden, D. C. and Graybill, F. A. (1966). Confidence bands of uniform and proportional width for linear models, Journal of the American Statistical Association 61, 182-198. Branco, M. D. and Dey, D. K. (2001). A general class of multivariate

References

249

skew-elliptical distributions, Journal of Multivariate Analysis 79, 99-113.

Bretz, F., Genz, A. and Hothorn, L. A. (2001). On the numerical availability of multiple comparison procedures, Biometrical Journal 43, 645-656. Bulgren, W. G. and Amos, D. E. (1968). A note on representation of the doubly non-central t distribution, Journal of the American Statistical Association 63, 1013-1019. Bulgren, W. G., Dykstra, R. L. and Hewett, J. E. (1974). A bivariate t distribution with applications, Journal of the American Statistical Association 69, 525-532. Cadwell, J. H. (1951). The bivariate normal integral, Biometrika 38, 475-481.

Cain, M. (1996). Forecasting with the maximum of correlated components having bivariate t-distributed errors, IMA Journal of Mathematics Applied in Business and Industry 7, 233-237. Capitanio, A., Azzalini, A. and Stanghellini, E. (2002). Graphical models for skew-normal variates, Scandinavian Journal of Statistics, to appear. Carlson, B. C. (1977). Special Functions and Applied Mathematics (Academic Press, New York). Castillo, E. and Sarabia, J. M. (1990). Bivariate distributions with second kind Beta conditionals, Communications in Statistics-Theory and Methods 19, 3433-3445. Chapman, D. G. (1950). Some two-sample tests, Annals of Mathematical Statistics 21, 601-606. Chen, H. J. (1979). Percentage points of multivariate t distribution with zero correlations and their application, Biometrical Journal 21, 347-360. Chib, S., Osiewalski, J. and Steel, M. F. J. (1991). Posterior inference on the degrees of freedom parameter in multivariate-t regression models, Economics Letters 37, 391-397. Chib, S., Tiwari, R. C. and Jammalamadaka, S. R. (1988). Bayes prediction in regressions with elliptical errors, Journal of Econometrics 38, 349-360.

Chien, J.-T. (2002). A Bayesian prediction approach to robust speech recognition and online environmental testing, Speech Communication 37, 321-334. Chow, Y. S. and Teicher, H. (1978). Probability Theory (Springer-Verlag, Berlin). Constantine, A. G. (1963). Some noncentral distribution problems in multivariate analysis, Annals of Mathematical Statistics 34, 1270-1285. Corliss, G. F. and Rall, L. B. (1987). Adaptive, self-validating numerical quadrature, SIAM Journal on Scientific and Statistical Computing 8, 831-847.

Cornish, E. A. (1954). The multivariate t distribution associated with a set of normal sample deviates, Australian Journal of Physics 7, 531-542. Cornish, E. A. (1955). The sampling distribution of statistics derived from the multivariate t distribution, Australian Journal of Physics 8, 193-199. Cornish, E. A. (1962). The multivariate t distribution associated with the general multivariate normal distribution, CSIRO Technical Paper No. 13, CSIRO Division in Mathematics and Statistics, Adelaide. Cornish, E. A. and Fisher, R. A. (1950). Moments and cumulants in the specification of distributions, in Contributions to Mathematical Statistics

250

References

(John Wiley and Sons, New York). Craig, C. (1936). On the frequency function of XY, Annals of Mathematical Statistics 7, 1-15. Cramer, H. (1951). Mathematical Methods of Statistics (Princeton University Press). DasGupta, A., Ghosh, J. K. and Zen, M. M. (1995). A new general method for constructing confidence sets in arbitrary dimensions: with applications, Annals of Statistics 23, 1408-1432. Dave, R. N. and Krishnapuram, R. (1995). Robust clustering methods: A unified view, IEEE Transactions on Fuzzy Systems 5, 270-293. David, H. A. (1982). Concomitants of order statistics: theory and applications, in Some Recent Advances in Statistics, ed. Tiago de Oliveira, pp. 89-100 (Academic Press, New York). Dawid, A. P. (1981). Some matrix-variate distribution theory: Notational considerations and a Bayesian application, Biometrika 68, 265-274. Dealt, I. (1990). Random Number Generators and Simulation (Akademiai Kiado, Budapest). Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion), Journal of the Royal Statistical Society B 39, 1-38. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1980). Iteratively weighted least squares for linear regression where errors are normal independent distributed, in Multivariate Analysis 5, ed. P. R. Krishnaiah, pp. 35-37 (North-Holland, New York). Dey, D. K. (1988). Simultaneous estimation of eigenvalues, Annals of the Institute of Statistical Mathematics 40, 137-147. Diaconis, P. and Ylvisaker, D. (1979). Conjugate priors for exponential families, Annals of Statistics 7, 269-281. Diaconis, P. and Ylvisaker, D. (1985). Quantifying prior opinion (with discussion), in Bayesian Statistics 2, ed. J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, pp. 133-156 (North Holland, Amsterdam). Dickey, J. M. (1965). Integrals of products of multivariate-t densities (abstract), Annals of Mathematical Statistics 36, 1611. Dickey, J. M. (1966a). Matric-variate generalizations of the multivariate t distribution and the inverted multivariate t distribution (abstract), Annals of Mathematical Statistics 37, 1423. Dickey, J. M. (1966b). On a multivariate generalization of the Behrens-Fisher distributions, Annals of Mathematical Statistics 37, 763. Dickey, J. M. (1967a). Expansions of t densities and related complete integrals, Annals of Mathematical Statistics 38, 503-510. Dickey, J. M. (1967b). Matric-variate generalizations of the multivariate t distribution and the inverted multivariate t distribution, Annals of Mathematical Statistics 38, 511-518. Dickey, J. M. (1968). Three multidimensional-integral identities with Bayesian applications, Annals of Mathematical Statistics 39, 1615-1627. Dickey, J. M., Dawid, A. and Kadane, J. B. (1986). Subjective-probability assessment methods for multivariate-t and matrix-t models, in Bayesian Inference and Decision Techniques, pp. 177-195 (North-Holland, Amsterdam). Dreier, I. and Kotz, S. (2002). A note on the characteristic function of the

References

251

t-distribution, Statistics and Probability Letters 57, 221-224. Dunn, O. J. (1958). Estimation of the means of dependent variables, Annals of Mathematical Statistics 29, 1095-1111. Dunn, O. J. (1961). Multiple comparison among means, Journal of the American Statistical Association 56, 52-64. Dunn, O. J. (1965). A property of the multivariate t distribution, Annals of Mathematical Statistics 36, 712-714. Dunn, O. J. and Massey, F. J. (1965). Estimation of multiple contrasts using t-distributions, Journal of the American Statistical Association 60, 573-583.

Dunnett, C. W. (1955). A multiple comparison procedure for comparing several treatments with a control, Journal of the American Statistical Association 50, 1096-1121. Dunnett, C. W. (1964). New tables for multiple comparisons with a control, Biometrics 20, 482-491. Dunnett, C. W. (1985). Multiple comparisons between several treatments and a specified treatment, in Linear Statistical Inference, Lecture Notes in Statistics No. 35, ed. T. Calinski and W. Klonecki, pp. 39-46 (Springer-Verlag, New York). Dunnett, C. W. (1989). Algorithm AS 251: Multivariate normal probability integrals with product correlation structure, Applied Statistics 38, 564-579.

Dunnett, C. W. and Sobel, M. (1954). A bivariate generalization of Student's t-distribution with tables for certain special cases, Biometrika 41, 153-169.

Dunnett, C. W. and Sobel, M. (1955). Approximations to the probability integral and certain percentage points of a multivariate analogue of Student's t-distribution, Biometrika 42, 258-260. Dunnett, C. W. and Tamhane, A. C. (1990). A step-up multiple test procedure, Technical Report 90-1, Department of Statistics, Northwestern University. Dunnett, C. W. and Tamhane, A. C. (1991). Step-down multiple tests for comparing treatments with a control in unbalanced one-way layouts, Statistics in Medicine 10, 939-947. Dunnett, C. W. and Tamhane, A. C. (1992). A step-up multiple test procedure, Journal of the American Statistical Association 87, 162-170. Dunnett, C. W. and Tamhane, A. C. (1995). Step-up multiple testing of parameters with unequally correlated estimates, Biometrics 51, 217-227. Dutt, J. E. (1973). A representation of multivariate normal probability integrals by integral transforms, Biometrika 60, 637-645. Dutt, J. E. (1975). On computing the probability integral of a general multivariate t, Biometrika 62, 201-205. Dutt, J. E., Mattes, K. D., Soms, A. P. and Tao, L. C. (1976). An approximation to the maximum modulus of the trivariate T with a comparison to exact values, Biometrics 32, 465-469. Dutt, J. E., Mattes, K. D. and Tao, L. C. (1975). Tables of the trivariate t for comparing three treatments to a control with unequal sample sizes, G. D. Searle and Company, Math. and Statist. Services, TR-3. Eaton, M. L. and Efron, B. (1970). Hotelling's T2 test under symmetry conditions, Journal of the American Statistical Association 65, 702-711. Edwards, D. E. and Berry, J. J. (1987). The efficiency of simulation based

252

References

multiple comparisons, Biometrics 43, 913-928. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1953). Higher Transcendental Functions, volumes 1 and 2 (McGraw-Hill, New York).

Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables with applications, Annals of Mathematical Statistics 38, 1466-1474.

Fang, H.-B., Fang, K.-T. and Kotz, S. (2002). The meta-elliptical distributions with given marginals, Journal of Multivariate Analysis 82, 1-16.

Fang, K.-T. and Anderson, T. W. (1990). Statistical Inference in Elliptically Contoured and Related Distributions (Alberton Press, New York). Fang, K.-T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions (Chapman and Hall, London). Fernandez, C. and Steel, M. F. J. (1999). Multivariate Student-t regression models: pitfalls and inference, Biometrika 86, 153-167. Fisher, R. A. (1925). Expansion of Student's integral in power of n-1, Metron 5, 109. Fisher, R. A. (1935). The fiducial argument in statistical inference, Annals of Eugenics 6, 391-398. Fisher, R. A. (1941). The asymptotic approach to Behren's integral with further tables for the d-test of significance, Ann. Eugen., Lond. 11, 141. Fisher, R. A. and Healy, M. J. R. (1956). New tables of Behrens' test of significance, Journal of the Royal Statistical Society B 18, 212-216. Fisher, R. A. and Yates, F. (1943). Statistical Tables for Biological, Agricultural and Medical Research, second edition (Oliver and Boyd, London). Fleishman, A. I. (1978). A method for simulating nonnormal distributions, Psychometrika 43, 521-532. Fraser, D. A. S. and Haq, M. S. (1969). Structural probability and prediction for the multivariate model, Journal of the Royal Statistical Society 31, 317-331.

Freeman, H. and Kuzmack, A. (1972). Tables of multivariate tin six or more dimensions, Biometrika 59, 217-219. Freeman, H., Kuzmack, A. and Maurice, R. (1967). Multivariate t and the ranking problem, Biometrika 54, 305-308. Frigui, H. and Krishnapuram, R. (1996). A robust algorithm for automatic extraction of an unknown number of clusters from noisy data, Pattern Recognition Letters 17, 1223-1232. Fry, R. L. (ed). (2002). Bayesian Inference and Maximum Entropy Methods in Science and Engineering: Proceedings of the 21st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (American Institute of Physics, New York). Fujikoshi, Y. (1987). Error bounds for asymptotic expansions of scale mixtures of distributions, Hiroshima Mathematics Journal 17, 309-324. Fujikoshi, Y. (1988). Non-uniform error bounds for asymptotic expansions of scale mixtures of distributions, Journal of Multivariate Analysis 27, 194-205.

Fujikoshi, Y. (1989). Error bounds for asymptotic expansions of the maximums of the multivariate t- and F-variables with common denominator, Hiroshima Mathematics Journal 19, 319-327.

References

253

Fujikoshi, Y. (1993). Error bounds for asymptotic approximations of some distribution functions, in Multivariate Analysis: Future Directions, ed. C. R. Rao, pp. 181-208 (North-Holland, Amsterdam). Fujikoshi, Y. (1997). An asymptotic expansion for the distribution of Hotelling's T2-statistic under nonnormality, Journal of Multivariate Analysis 61, 187-193. Fujikoshi, Y. and Shimizu, R. (1989). Error bounds for asymptotic expansions of scale mixtures of univariate and multivariate distributions, Journal of Multivariate Analysis 30, 279-291. Fujikoshi, Y. and Shimizu, R. (1990). Asymptotic expansions of some distributions and their error bounds-the distributions of sums of independent random variables and scale mixtures, Sugaku Expositions 3, 75-96.

Geisser, S. (1965). Bayesian estimation in multivariate analysis, Annals of Mathematical Statistics 36, 150-159. Geisser, S. and Cornfield, J. (1963). Posterior distributions for multivariate normal parameters, Journal of the Royal Statistical Society B 25, 368-376.

Genz, A. (1992). Numerical computation of the multivariate normal probabilities, Journal of Computational and Graphical Statistics 1, 141-150.

Genz, A. and Bretz, F. (1999). Numerical computation of multivariate t probabilities with application to power calculation of multiple contrasts, Journal of Statistical Computation and Simulation 63, 361-378. Genz, A. and Bretz, F. (2001). Methods for the computation of multivariate t-probabilities, Journal of Computational and Graphical Statistics. Ghosh, B. K. (1973). Some monotonicity theorems for x2, F and t distributions with applications, Journal of the Royal Statistical Society B 35, 480-492. Ghosh, B. K. (1975). On the distribution of the difference of two t variables, Journal of the American Statistical Association 70, 463-467. Gill, M. L., Tiku, M. L. and Vaughan, D. C. (1990). Inference problems in life testing under multivariate normality, Journal of Applied Statistics 17, 133-147. Glaz, J. and Johnson, B. McK. (1984). Probability inequalities for multivariate distributions with dependence structures, Journal of the American Statistical Association 79, 435-440. Goldberg, H. and Levine, H. (1946). Approximate formulas for the percentage points and normalization of t and x2, Annals of Mathematical Statistics 17, 216. Goodman, M. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction), Annals of Mathematical Statistics 34. Graybill, F. A. and Bowden, D. C. (1967). Linear segment confidence bands for simple linear models, Journal of the American Statistical Association 62, 403-408. Grosswald, E. (1976). The Student t-distribution of any degree of freedom is infinitely divisible, Zeitschrift fur Wahrscheinlichkeitstheorie and Verwandte Gebiete 36, 103-109. Guerrero-Cusumano, J.-L. (1996a). A measure of total variability for the multivariate t distribution with applications to finance, Information

254

References

Sciences 92, 47-63. Guerrero-Cusumano, J.-L. (1996b). An asymptotic test of independence for multivariate t and Cauchy random variables with applications, Information Sciences 92, 33-45. Guerrero-Cusumano, J.-L. (1998). Measures of dependence for the multivariate t distribution with applications to the stock market, Communications in Statistics-Theory and Methods 27, 2985-3006. Gupta, A. K. (2000). Multivariate skew t distribution, Technical Report No. 00-04, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio. Gupta, A. K. and Kollo. T. (2000). Multivariate skew normal distribution: some properties and density expansions, Technical Report No. 00-03, Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio. Gupta, R. P. (1964). Some extensions of the Wishart and multivariate t distributions in the complex case, Journal of the Indian Statistical Association 2, 131-136. Gupta, S. S. (1963). Probability integrals of multivariate normal and multivariate t, Annals of Mathematical Statistics 34, 792-828. Gupta, S. S., Nagel, K. and Panchapakesan, S. (1973). On the order statistics from equally correlated normal random variables, Biometrika 60, 403-413. Gupta, S. S., Panchapakesan, S. and Sohn, J. K. (1985). On the distribution of the studentized maximum of equally correlated normal random variables, Communications in Statistics-Simulation and Computation 14, 103-135. Gupta, S. S. and Sobel, M. (1957). On a statistic which arises in selection and ranking problems, Annals of Mathematical Statistics 28, 957-967. Hahn, G. J. and Hendrickson, R. W. (1971). A table of percentage points of the distribution of the largest absolute value of k Student t variates and its application, Biometrika 58, 323-332. Hahn, M. G. and Klass, M. J. (1980a). Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal, Annals of Probability 8, 262-280. Hahn, M. G. and Klass, M. J. (1980b). The generalized domain of attraction of spherically symmetric stable laws in ltd, in Proceedings of the Conference on Probability in Vector Spaces II, Lecture Notes in Mathematics 828, pp. 52-81 (Springer-Verlag, New York). Halgreen, C. (1979). Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Zeitschrift fur Wahrscheinlichkeitstheorie and Verwandte Gebiete 47, 13-17. Hall, P. (1989). On polynomial-based projection indices for exploratory projection pursuit, Annals of Statistics 17, 589-605. Halperin, M. (1967). An inequality on a bivariate t distribution, Journal of the American Statistical Association 62, 603-606. Halperin, M., Greenhouse, S. W., Cornfield, J. and Zalokar, J. (1955). Tables of percentage points for the studentized maximum absolute deviate in normal samples, Journal of the American Statistical Association 50, 185-195.

Hammersley, J. M. and Handscomb, D. C. (1964). Monte Carlo Methods (Methuen & Co. Ltd, London).

References

255

Haq, M. S. and Khan, S. (1990). Prediction distribution for a linear regression model with multivariate Student-t error distribution, Communications in Statistics-Theory and Methods 19, 4705-4712. Harter, H. L. (1951). On the distribution of Wald's classification statistic, Annals of Mathematical Statistics 22, 58-67. Hayakawa, T. (1989). On the distributions of the functions of the F-matrix under an elliptical population, Journal of Statistical Planning and Inference 21, 41-52. Hochberg, Y. and Tambane, A. C. (1987). Multiple Comparison Procedures (John Wiley and Sons, New York). Hsu, H. (1990). Noncentral distributions of quadratic forms for elliptically contoured distributions, in Statistical Inference in Elliptically Contoured and Related Distributions, pp. 97-102 (Allerton, New York). Hsu, J. C. (1992). The factor analytic approach to simultaneous inference in the general linear model, Journal of Computational and Graphical Statistics 1, 151-168. Hsu, J. C. and Nelson, B. L. (1998). Multiple comparisons in the general linear model, Journal of Computational and Graphical Statistics 7, 23-41.

Hutchinson, T. P. and Lai, C. D. (1990). Continuous Bivariate Distributions, Emphasising Applications (Rumsby, Adelaide). Ifram, A. F. (1970). On the characteristic function of F and t distributions, Sankhya A 32, 350-352. International Mathematical and Statistical Libraries (1987). MATH/Library, Fortran Subroutines for Mathematical Applications (International Mathematical and Statistical Libraries, Houston). Iwashita, T. (1997). Asymptotic null and nonnull distribution of Hotelling's T2-statistic under the elliptical distribution, Journal of Statistical Planning and Inference 61, 85-104. Iyengar, S. (1988). Evaluation of normal probabilities of symmetric regions, SIAM Journal on Scientific and Statistical Computing 9, 418-423. James, A. T. (1964). Distribution of matrix variates and latent roots derived from normal samples, Annals of Mathematical Statistics 35, 475-501. James, W. and Stein, C. (1961). Estimation with quadratic loss, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability 1, pp. 361-379. Javier, W. R. and Gupta, A. K. (1985). On matric variate-t distribution, Communications in Statistics-Theory and Methods 14, 1413-1425. Javier, W. R. and Srivastava, T. N. (1988). On the multivariate t distribution, Pakistan Journal of Statistics 4, 101-109. Jaynes, E. T. (1957). Information theory and statistical mechanics, Physics Review 106, 620-630. Jensen, D. R. (1994). Closure of multivariate t and related distributions, Statistics and Probability Letters 20, 307-312. Joarder, A. H. (1995). Estimation of the trace of the scale matrix of a multivariate t-model, in Proceedings of the Econometrics Conference, pp. 467-474 (Monash University, Australia). Joarder, A. H. (1998). Some useful Wishart expectations based on the multivariate t-model, Statistical Papers 39, 223-229. Joarder, A. H. and Ahmed, S. E. (1996). Estimation of the characteristic roots of the scale matrix, Metrika 44, 259-267.

References

256

Joarder, A. H. and Ahmed, S. E. (1998). Estimation of the scale matrix of a class of elliptical distributions, Metrika 48, 149-160. Joarder, A. H. and Ali, M. M. (1992). On some generalized Wishart expectations, Communications in Statistics-Theory and Methods 21, 283-294.

Joarder, A. H. and Ali, M. M. (1996). On the characteristic function of the multivariate t distribution, Pakistan Journal of Statistics 12, 55-62. Joarder, A. H. and Ali, M. M. (1997). Estimation of the scale matrix of a multivariate t-model under entropy loss, Metrika 46, 21-32. Joarder, A. H. and Singh, S. (1997). Estimation of the trace of the scale matrix of a multivariate t-model using regression type estimator, Statistics 29, 161-168. Joe, H. (1989). Relative entropy measures of multivariate dependence, Journal of the American Statistical Association 84, 157-164. Joe, S. (1990). Randomization of lattice rules for numerical multiple integration, Journal of Computational and Applied Mathematics 31, 299-304.

Jogdeo, K. (1977). Association of probability inequalities, Annals of Statistics 5, 495-504. John, S. (1961). On the evaluation of the probability integral of the multivariate t distribution, Biometrika 48, 409-417. John, S. (1964). Methods for the evaluation of probabilities of polygonal and angular regions when the distribution is bivariate t, Sankhya A 26, 47-54.

John, S. (1966). On the evaluation of probabilities of convex polyhedra under multivariate normal and t distributions, Journal of the Royal Statistical Society B 28, 366-369. Johnson, M. (1987). Multivariate Statistical Simulation (John Wiley and Sons, New York). Johnson, N. L. and Kotz, S. (1972). Distributions in Statistics: Continuous Multivariate Distributions (John Wiley and Sons, New York). Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, volume 2, second edition (John Wiley and Sons, New York). Johnson, R. A. and Weerahandi, S. (1988). A Bayesian solution to the multivariate Behrens-Fisher problem, Journal of the American Statistical Association 83, 145-149. Jolion, J.-M., Meer, P. and Bataouche, S. (1995). Robust clustering with applications in computer vision, IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 791-802. Jones, M. C. (2001a). A skew t distribution, in Probability and Statistical Models with Applications, ed. C. A. Charalambides, M. V. Koutras and N. Balakrishnan, pp. 269-278 (Chapman and Hall, London). Jones, M. C. (2001b). Multivariate t and beta distributions associated with multivariate F distribution, Metrika 54, 215-231. Jones, M. C. (2002a). A bivariate distribution with support above the diagonal and skew t marginals. Submitted. Jones, M. C. (2002b). A dependent bivariate t distribution with marginal on different degrees of freedom, Statistics and Probability Letters 56, 163-170.

Jones, M. C. (2002c). Marginal replacement in multivariate densities, with

References

257

application to skewing spherically symmetric distributions, Journal of Multivariate Analysis 81, 85-99. Jones, M. C. and Faddy, M. J. (2002). A skew extension of the t distribution with applications. Submitted. Jones, M. C. and Sibson, R. (1987). What is projection pursuit (with discussion)? Journal of the Royal Statistical Society A 150, 1-36. Kabe, D. G. and Gupta, A. K. (1990). Hotelling's T2-distribution for a mixture of two normal populations, South African Statistical Journal 24, 87-92. Kano, Y. (1994). Consistency property of elliptical probability density functions, Journal of Multivariate Analysis 51, 139-147. Kano, Y. (1995). An asymptotic expansion of the distribution of Hotelling's T2-statistic under general distributions, American Journal of Mathematical and Management Sciences 15, 317-341. Kappenman, R. F. (1971). A note on the multivariate t ratio distribution, Annals of Mathematical Statistics 42, 349-351. Kass, R. E. and Steffey, D. (1989). Approximate Bayesian in conditionally independent hierarchical models, Journal of the American Statistical Association 84, 717-726. Kelejian, H. H. and Prucha, I. R. (1985). Independent or uncorrelated disturbances in linear regression: An illustration of the difference, Economic Letters 19, 35-38. Kelker, D. (1970). Distribution theory of spherical distributions and location scale parameters, Sankhya A 32, 419-430. Kelker, D. (1971). Infinite divisibility and variance mixtures of the normal distribution, Annals of Mathematical Statistics 42, 802-808. Kendall, M. G. and Stuart, A. (1958). The Advanced Theory of Statistics (Hafner, New York). Kent, J. T., Tyler, D. E. and Vardi, Y. (1994). A curious likelihood identity for the multivariate t-distribution, Communications in Statistics-Simulation and Computation 23, 441-453. Kharin, Y. (1996). Robustness in Statistical Pattern Recognition (Kluwer, Dordrecht). Khatri, C. G. (1967). On certain inequalities for normal distributions and their applications to simultaneous confidence bands, Annals of Mathematical Statistics 38, 1853-1867. Kiefer, J. and Schwarz, R. (1965). Admissible Bayes character of T2-, R2-, and other fully invariant tests for classical multivariate normal problems, Annals of Mathematical Statistics 36, 747-770. Kopal, Z. (1955). Numerical Analysis (Chapman and Hall, London). Kottas, A., Adamidis, K. and Loukas, S. (1999). Bvariate distributions with Pearson type VII conditionals, Annals of the Institute of Statistical Mathematics 51, 331-344. Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000). Continuous Multivariate Distributions, Volume 1: Models and Applications, second edition (John Wiley and Sons, New York). Kotz, S., Lumelski, Y. and Pensky, M. (2003). Strength Stress Models with Applications (World Scientific Press, Singapore). Kozumi, H. (1994). Testing equality of the means in two independent multivariate t distributions, Communications in Statistics-Theory and Methods 23, 215-227.

References

258

Krishnaiah, P. R. and Armitage, J. V. (1966). Tables for multivariate t distribution, Sankhya B 28, 31-56. Krishnan, M. (1959). Studies in statistical inference, Ph.D. Thesis, Madras University, India. Krishnan, M. (1967a). The moments of a doubly noncentral t distribution, Journal of the American Statistical Association 62, 278-287. Krishnan, M. (1967b). The noncentral bivariate chi distribution, SIAM Review 9, 708-714. Krishnan, M. (1968). Series representations of the doubly noncentral t distribution, Journal of the American Statistical Association 63, 1004-1012.

Krishnan, M. (1970). The bivariate doubly noncentral t distribution (abstract), Annals of Mathematical Statistics 41, 1135. Krishnan, M. (1972). Series representations of a bivariate singly noncentral t distribution, Journal of the American Statistical Association 67, 228-231.

Kshirsagar, A. M. (1961). Some extensions of the multivariate generalization t distribution and the multivariate generalization of the distribution of the regression coefficient, in Proceedings of the Cambridge Philosophical Society 57, pp. 80-85. Kudo, A. (1963). A multivariate analogue of the one-sided test, Biometrika 50, 403-418. Kullback, S. (1968). Information Theory and Statistics (John Wiley and Sons, New York). Kunte, S. and Rattihalli, R. N. (1984). Rectangular regions of maximum probability content, Annals of Statistics 12, 1106-1108. Kurths, J., Voss, A. and Saparin, P., Witt, A., Kleiner, H. J. and Wessel, N. (1995). Quantitative analysis of heart rate variability, Chaos 1, 88-94. Kwong, K.-S. (2001a). A modified Dunnett and Tamhane step-up approach for establishing superiority/equivalence of a new treatment compared with k standard treatments, Journal of Statistical Planning and Inference 97, 359-366. Kwong, K.-S. (2001b). An algorithm for construction of multiple hypothesis testing, Computational Statistics 16, 165-171. Kwong, K.-S. and Iglewicz, B. (1996). On singular multivariate normal distribution and its applications, Computational Statistics and Data Analysis 22, 271-285. Kwong, K.-S. and Liu, W. (2000). Calculation of critical values for Dunnett and Tamhane's step-up multiple test procedure, Statistics and Probability Letters 49, 411-416. Landenna, G. and Ferrari, P. (1988). The k-variate student distribution and a test with the control of type I error in multiple decision problems, Technical Report, Istituto di Scienze Statistiche e Matematiche, University di Milano, Italy. Lange, K. and Sinsheimer, J. S. (1993). Normal/independent distributions and their applications in robust regression, Journal of Computational and Graphical Statistics 2, 175-198. Lange, K. L., Little, R. J. A. and Taylor, J. M. G. (1989). Robust statistical modeling using the t distribution, Journal of the American Statistical Association 84, 881-896. Lauprete, G. J., Samarov, A. M. and Welsch, R. E. (2002). Robust portfolio

References

259

optimization, Metrika 55, 139-149. Lazo, A. and Rathie, P. (1978). On the entropy of continuous probability distributions, IEEE Transactions on Information Theory 24, 120-122. Lebedev, N. N. (1965). Special Functions and Their Applications (Prentice-Hall Inc., New Jersey). Lee, R. E. and Spurrier, J. D. (1995). Successive comparisons between ordered treatments, Journal of Statistical Planning and Inference 43, 323-330.

Lehmann, E. L. (1966). Some concepts of dependence, Annals of Mathematical Statistics 37, 1137-1153. Leonard, T. (1982). Comment on "A simple predictive density function" by M. Lejeune and G. D. Faukkenberry," Journal of the American Statistical Association 77, 657-658. Leonard, T., Hsu, J. S. J. and Ritter, C. (1994). The Laplacian T-approximation in Bayesian inference, Statistica Sinica 4, 127-142. Leonard, T., Hsu, J. S. J. and Tsui, K. W. (1989). Bayesian marginal inference, Journal of the American Statistical Association 84, 1051-1058.

Lin, P. (1972). Some characterizations of the multivariate t distribution, Journal of Multivariate Analysis 2, 339-344. Linfoot, E. (1957). An informational measure of correlation, Information and Control 1, 85-89. Little, R. J. A. (1988). Robust estimation of the mean and covariance matrix from data with missing values, Applied Statistics 37, 23-39. Little, R. J. A. and Rubin, D. B. (1987). Statistical Analysis with Missing Data (John Wiley and Sons, New York). Liu, C. (1993). Bartlett's decomposition of the posterior distribution of the covariance for normal monotone ignorable missing data, Journal of Multivariate Analysis 46, 198-206. Liu, C. (1995). Missing data imputation using the multivariate t distribution, Journal of Multivariate Analysis 53, 139-158. Liu, C. (1996). Bayesian robust multivariate linear regression with incomplete data, Journal of the American Statistical Association 91, 1219-1227.

Liu, C. (1997). ML estimation of the multivariate t distribution and the EM algorithm, Journal of Multivariate Analysis 63, 296-312. Liu, C. and Rubin, D. B. (1995). ML estimation of the multivariate t distribution with unknown degrees of freedom, Statistica Sinica 5, 19-39. Liu, C., Rubin, D. B. and Wu, Y. N. (1998). Parameter expansion to accelerate EM: the PX-EM algorithm, Biometrika 85, 755-770. Liu, W., Miwa, T. and Hayter, A. J. (2000). Simultaneous confidence interval estimation for successive comparisons of ordered treatment effects, Journal of Statistical Planning and Inference 88, 75-86. Magnus, J. R. and Neudecker, H. (1979). The commutation matrix: Some properties and applications, Annals of Statistics 7, 381-394. Magnus, W., Oberhettinger, F. and Soni, R. P. (1966). Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York).

Mann, N. R. (1982). Optimal outlier tests for a Weibull model - To identify process changes or to predict failure times, TIMS/Studies in the Management Sciences 19, 261-279.

260

References

Mardia, K. V. (1970a). Families of Bivariate Distributions (Griffin, London). Mardia, K. V. (1970b). Measures of multivariate skewness and kurtosis with applications, Biometrika 57, 519-530. Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter, Annals of Statistics 4, 51-67. Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables, Journal of the American Statistical Association 60, 193-204.

Marshall, A. W. and Olkin, I. (1974). Majorization in multivariate distributions, Annals of Statistics 2, 1189-1200. McLachlan, G. J. and Peel, D. (1998). Robust cluster analysis via mixtures of multivariate t-distributions, in Lecture Notes in Computer Science 1451, ed. A. Amin, D. Dori, P. Pudil amd H. Freeman, pp. 658-666 (Springer-Verlag, Berlin). McLachlan, G. J., Peel, D., Basford, K. E. and Adams, P. (1999). Fitting of mixtures of normal and t components, Journal of Statistical Software 4. Meng, X. L. and van Dyk, D. (1997). The EM algorithm-An old folk song sung to fast new tune (with discussion), Journal of the Royal Statistical Society B 59, 511-567. McCann, M. and Edwards, D. (1996). A path inequality for the multivariate t distribution, with applications to multiple comparisons, Journal of the American Statistical Association 91, 211-216. Miller, K. S. (1968). Some multivariate t distributions, Annals of Mathematical Statistics 39, 1605-1609. Milton, R. C. (1963). Tables of the equally correlated multivariate normal probability integral, Technical Report No. 27, University of Minnesota, Minneapolis.

Morales, D., Pardo, L. and Vajda, I. (1997). Some new statistics for testing hypotheses in parametric models, Journal of Multivariate Analysis 62, 137-168.

Nagarsenker, B. N. (1975). Some distribution problems connected with multivariate t distribution, Metron 33, 66-74. Nason, G. P. (2000). Analytic formulae for projection indices in a robustness experiment, Technical Report 00:06, Department of Mathematics, University of Bristol. Nason, G. P. (2001). Robust projection indices, Journal of the Royal Statistical Society B 63, 551-567. Neyman, J. (1959). Optimal asymptotic tests for composite hypotheses, in Probability and Statistics, ed. U. Grenander, pp. 213-234 (John Wiley and Sons, New York). Nicholson, C. (1943). The probability integral for two variables, Biometrika 33, 59-72. Osiewalski, J. (1991). A note on Bayesian inference in a regression model with elliptical errors, Journal of Econometrics 48, 183-193. Osiewalski, J. and Steel, M. F. J. (1990). Robust Bayesian inference in elliptical regression models, Center Discussion Paper 9032, Tilburg University.

Owen, D. B. (1956). Tables for computing bivariate normal probabilities, Annals of Mathematical Statistics 27, 1075-1090. Owen, D. B. (1965). A special case of a bivariate non-central t distribution, Biometrika 52, 437-446.

References

261

Patil, S. A. and Kovner, J. L. (1968). On the probability of trivariate Student's t distribution (abstract), Annals of Mathematical Statistics 39, 1784.

Patil, S. A. and Kovner, J. L. (1969). On the bivariate doubly noncentral t distributions (abstract), Annals of Mathematical Statistics 40, 1868. Patil, S. A. and Liao, S. H. (1970). The distribution of the ratios of means to the square root of the sum of variances of a bivariate normal sample, Annals of Mathematical Statistics 41, 723-728. Patil, V. H. (1965). Approximation to the Behrens-Fisher distributions, Biometrika 52, 267-271. Patnaik, P. B. (1955). Hypotheses concerning the means of observations in normal samples, Sankhya 15, 343-372. Paulson, E. (1952). On the comparison of several experimental categories with a control, Annals of Mathematical Statistics 23, 239-246. Pearson, K. (1923). On non-skew frequency surfaces, Biometrika 15, 231. Pearson, K. (1931). Tables for Statisticians and Biometricians, Part II (Cambridge University Press for the Biometrika Trust, London). Peel, D. and McLachlan, G. J. (2000). Robust mixture modelling using the t distribution, Statistics and Computing 10, 339-348. Pestana, D. (1977). Note on a paper of Ifram, Sankhya A 39, 396-397. Pillai, K. C. S. and Ramachandran, K. V. (1954). Distribution of a Studentized order statistic, Annals of Mathematical Statistics 25, 565-571.

Press, S. J. (1969). The t ratio distribution, Journal of the American Statistical Association 64, 242-252. Press, S. J. (1972). Applied Multivariate Analysis (Holt, Rinehart and Winston, Inc, New York). Press, W. H. (1986). Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, Cambridge). Raiffa, H. and Schlaifer, R. (1961). Applied Statistical Decision Theory (Harvard University Press, Cambridge, MA). Rattihalli, R. N. (1981). Regions of maximum probability content and their applications, Ph.D. Thesis, University of Poona, India. Rausch, W. and Horn, M. (1988). Applications and tabulations of the multivariate t distribution with p = 0, Biometrical Journal 30, 595-605. Renyi, A. (1959). On the dimension and entropy of probability distributions, Acta Mathematica Academiae Scientiarum Hungaricae 10, 193-215. Renyi, A. (1960). A few fundamental problems of information theory (in Hungarian), A Magyar Tudomanyos Akademia Matematikai es Fizikai Tudomanyok Osztalyanak Kozlemenyei 10, 251-282. Renyi, A. (1961). On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability I, pp. 547-561 (University of California Press, Berkeley). Robbins, H. (1948). The distribution of Student's t when the population means are unequal, Annals of Mathematical Statistics 19, 406-410. Rousseeuw, P. J., Kaufman, L. and Trauwaert, E. (1996). Fuzzy clustering using scatter matrices, Computational Statistics and Data Analysis 23, 135-151.

Ruben, H. (1960). On the distribution of weighted difference of two independent Student variates, Journal of the Royal Statistical Society B 22, 188-194.

References

262

Rubin, D. B. (1983). Iteratively reweighted least squares, in Encyclopedia of Statistical Sciences 4, ed. S. Kotz and N. L. Johnson, pp. 272-275 (John Wiley and Sons, New York). Rubin, D. B. (1987). Multiple Imputation for Nonresponse in Surveys (John Wiley and Sons, New York). Rubin, D. B. and Schafer, J. L. (1990). Efficiently creating multiple imputations for incomplete multivariate normal data, in Proceedings of the Statistical Computing Section of the American Statistical Association, pp. 83-88 (American Statistical Association, Washington, DC). Sahu, S. K., Dey, D. K. and Branco, M. D. (2000). A new class of multivariate skew distributions with applications to Bayesian regression models, Research Report RT-MAE 2000-16, Department of Statistics, University of Sao Paulo, Sao Paulo, Brasil. Sarabia, J. M. (1995). The centered normal conditionals distribution, Communications in Statistics-Theory and Methods 24, 2889-2900. Schafer, J. L. (1997). Analysis of Incomplete Multivariate Data (Chapman and Hall, London). Scott, A. (1967). A note on conservative confidence regions for the mean of a multivariate normal, Annals of Mathematical Statistics 38, 278-280. Correction: Annals of Mathematical Statistics 39, 1968, 2161. Seal, K. C. (1954). On a class of decision procedures for ranking means, Institute of Statistics Mimeograph Series No. 109, University of North Carolina at Chapel Hill. Seneta, E. (1993). Probability inequalities and Dunnett's test, in Multiple Comparisons, Selection, and Applications in Biometry, pp. 29-45 (Marcel Dekker, New York). Sepanski, S. J. (1994). Asymptotics for multivariate t-statistic and Hotelling's T2-statistic under infinite second moments via bootstrapping, Journal of Multivariate Analysis 49, 41-54. Sepanski, S. J. (1996). Asymptotics for multivariate t-statistic for random vectors in the generalized domain of attraction of the multivariate normal law, Statistics and Probability Letters 30, 179-188. Shampine, L. F. and Allen, R. C. (1973). Numerical Computing: An Introduction (Saunders, Philadelphia). Shimizu, R. and Fujikoshi, Y. (1997). Sharp error bounds for asymptotic expansions of the distribution functions for scale mixtures, Annals of the Institute of Statistical Mathematics 49, 285-297. Sidak, Z. (1965). Rectangular confidence regions for means of multivariate normal distributions, Bulletin of the Institute of International Statistics 41, 380-381. Sidak, Z. (1967). Rectangular confidence regions for the means of multivariate normal distributions, Journal of the American Statistical Association 62, 626-633. Sidak, Z. (1971). On probabilities of rectangles in multivariate Student distributions: their dependence and correlations, Annals of Mathematical Statistics 42, 169-175. Sidak, Z. (1973). A chain of inequalities for some types of multivariate distributions, with nine special cases, Applications of Mathematics 18, 110-118.

Siddiqui, M. M. (1967). A bivariate t distribution, Annals of Mathematical

References

263

Statistics 38, 162-166. Singh, R. K. (1988). Estimation of error variance in linear regression models with errors having multivariate student-t distribution with unknown degrees of freedom, Economics Letters 27, 47-53. Singh, R. K. (1991). James-Stein rule estimators in linear regression models with multivariate-t distributed error, Australian Journal of Statistics 33, 145-158.

Singh, R. K., Mistra, S. and Pandey, S. K. (1995). A generalized class of estimators in linear regression models with multivariate-t distributed error, Statistics and Probability Letters 23, 171-178. Siotani, M. (1959). The extreme value of the generalized distances of the individual points in the multivariate normal sample, Annals of the Institute of Statistical Mathematics 10, 183-208. Siotani, M. (1964). Interval estimation for linear combinations of means, Journal of the American Statistical Association 59, 1141-1164. Siotani, M. (1976). Conditional and stepwise multivariate t distributions, in Essays in Probability and Statistics, pp. 287-303 (Tokyo). Singh, R. S. (1991). James-Stein rule estimators in linear regression models with multivariate-t distributed error, Australian Journal of Statistics 33, 145-158.

Sloan, I. H. and Joe, S. (1994). Lattice Methods for Multiple Integration (Clarendon Press, Oxford). Smith, D. J., Bailey, T. C. and Munford, G. (1993). Robust classification of high-dimensional data using artificial neural networks, Statistics and Computing 3, 71-81. Somerville, P. N. (1993a). Simultaneous confidence intervals (General linear model), Bulletin of the International Statistical Institute 2, 427-428. Somerville, P. N. (1993b). Exact all-pairwise multiple comparisons for the general linear model, in Proceedings of the 25th Symposium on the Interface, Computing Science and Statistics, pp. 352-356 (Interface Foundation, Virginia) Somerville, P. N. (1993c). Simultaneous multiple orderings, Technical Report TR-93-1, Department of Statistics, University of Central Florida, Orlando. Somerville, P. N. (1994). Multiple comparisons, Technical Report TR-94-1, Department of Statistics, University of Central Florida, Orlando. Somerville, P. N. (1997). Multiple testing and simultaneous confidence intervals: Calculation of constants, Computational Statistics and Data Analysis 25, 217-233. Somerville, P. N. (1998a). A Fortran 90 program for evaluation of multivariate normal and multivariate-t integrals over convex regions, Journal of Statistical Software, http://www.stat.ucla.edu/journals/jss/v03/i04. Somerville, P. N. (1998b). Numerical computation of multivariate normal and multivariate t probabilities over convex regions, Journal of Computational and Graphical Statistics 7, 529-544. Somerville, P. N. (1999a). Numerical evaluation of multivariate integrals over ellipsoidal regions, Bulletin of the International Statistical Institute. Somerville, P. N. (1999b). Critical values for multiple testing and comparisons: one step and step down procedures, Journal of Statistical Planning and Inference 82, 129-138.

References

264

Somerville, P. N. (2001). Numerical computation of multivariate normal and multivariate t probabilities over ellipsoidal regions, Journal of Statistical Software,

http: //www. stat.ucla.edu/www. jstatsoft. org/v06 /iO8.

Somerville, P. N. and Bretz, F. (2001). Fortran 90 and SAS-IML programs for computation of critical values for multiple testing and simultaneous confidence intervals, Journal of Statistical Software, http://www.stat.ucla.edu/www.jstatsoft.org/v06/iO5. Somerville, P. N., Miwa, T., Liu, W. and Hayter, A. (2001). Combining one-sided and two-sided confidence interval procedures for successive comparisons of ordered treatment effects, Biometrical Journal 43, 533-542.

Song, K.-S. (2001). Renyi information, loglikelihood and an intrinsic distribution measure, Journal of Statistical Planning and Inference 93, 51-69.

Spainer, J. and Oldham, K. B. (1987). An Atlas of Functions (Hemisphere Publishing Company, Washington, DC). Spurrier, J. D. and Isham, S. P. (1985). Exact simultaneous confidence intervals for pairwise comparisons of three normal means, Journal of the American Statistical Association 80, 438-442. Srivastava, M. S. and Awan, H. M. (1982). On the robustness of Hotelling's T2-test and distribution of linear and quadratic forms in sampling from a mixture of two multivariate normal populations, Communications in Statistics-Theory and Methods 11, 81-107. Steffens, F. E. (1969a). A stepwise multivariate t distribution, South African Statistical Journal 3, 17-26. Steffens, F. E. (1969b). Critical values for bivariate Student t-tests, Journal of the American Statistical Association 64, 637-646. Steffens, F. E. (1970). Power of bivariate studentized maximum and minimum modulus tests, Journal of the American Statistical Association 65, 1639-1644. Steffens, F. E. (1974). A bivariate t distribution which occurs in stepwise regression (abstract), Biometrics 30, 385. Steyn, H. S. (1993). One the problem of more than one kurtosis parameter in multivariate analysis, Journal of Multivariate Analysis 44, 1-22. Stone, M. (1964). Comments on a posterior distribution of Geisser and Cornfield, Journal of the Royal Statistical Society B 26, 274-276. Sukhatme, P. V. (1938). On Fisher and Behrens' test of significance for the difference in means of two normal samples, Sankhya 4, 39-48. Sultan, S. A. and Tracy, D. S. (1996). Moments of the complex multivariate normal distribution. Special issue honoring Calyampudi Radhakrishna Rao, Linear Algebra and Its Applications 237/238, 191-204. Sun, L., Hsu, J. S. J., Guttman, I. and Leonard, T. (1996). Bayesian methods for variance component models, Journal of the American Statistical Association 91, 743-752. Sutradhar, B. C. (1986). On the characteristic function of the multivariate Student t-distribution, Canadian Journal of Statistics 14, 329-337. Sutradhar, B. C. (1988a). Author's revision, Canadian Journal of Statistics 16, 323. Sutradhar, B. C. (1988b). Testing linear hypothesis with t error variable, Sankhya B 175-180.

References

265

Sutradhar, B. C. (1990). Discrimination of observations into one of two t populations, Biometrics 46, 827-835. Sutradhar, B. C. (1993). Score test for the covariance matrix of the elliptical t-distribution, Journal of Multivariate Analysis 46, 1-12. Sutradhar, B. C. and Ali, M. M. (1986). Estimation of the parameters of a regression model with a multivariate t error variable, Communications in Statistics-Theory and Methods 15, 429-450. Sutradhar, B. C. and Ali, M. M. (1989). A generalization of the Wishart distribution for the elliptical model and its moments for the multivariate t model, Journal of Multivariate Analysis 29, 155-162. Sweeting, T. J. (1984). Approximate inference in location-scale regression models, Journal of the American Statistical Association 79, 847-852. Sweeting, T. J. (1987). Approximate Bayesian analysis of censored survival data, Biometrika 74, 809-816. Takano, K. (1994). On Bessel equations and the Levy representation of the multivariate t distribution, Technical Report, Department of Mathematics, Ibaraki University, Japan. Tan, W. Y. (1969a). Note on the multivariate and the generalized multivariate beta distributions, Journal of the American Statistical Association 64, 230-241. Tan, W. Y. (1969b). Some distribution theory associated with complex Gaussian distribution, Tamkang Journal 7, 263-302. Tan, W. Y. (1973). On the complex analogue of Bayesian estimation of a multivariate regression model, Annals of the Institute of Statistical Mathematics 25, 135-152. Tiao, G. C. and Zellner, A. (1964). On the Bayesian estimation of multivariate regression, Journal of the Royal Statistical Society B 26, 277-285.

Tierney, L. and Kadane, J. (1986). Accurate approximations for posterior moments and marginal densities, Journal of the American Statistical Association 81, 82-86. Tiku, M. L. (1967). Tables of the power of the F-test, Journal of the American Statistical Association 62, 525-539. Tiku, M. L. and Gill, P. S. (1989). Modified maximum likelihood estimators for the bivariate normal based on Type II censored samples, Communications in Statistics-Theory and Methods 18, 3505-3518. Tiku, M. L. and Kambo, N. S. (1992). Estimation and hypothesis testing for a new family of bivariate nonnormal distributions, Communications in Statistics-Theory and Methods 21, 1683-1705. Tiku, M. L. and Suresh, R. P. (1992). A new method of estimation for location and scale parameters, Journal of Statistical Planning and Inference 30, 281-292. Tong, Y. L. (1970). Some probability inequalities of multivariate normal and multivariate t, Journal of the American Statistical Association 65, 1243-1247.

Tong, Y. L. (1982). Rectangular and elliptical probability inequalities for Schur-concave random variables, Annals of Statistics 10, 637-642. Tranter, C. J. (1968). Bessel Functions with Some Physical Applications (English Universities Press Ltd., London). Trout, J. R. and Chow, B. (1972). Table of the percentage points of the trivariate t distribution with an application to uniform confidence

References

266

bands, Technometrics 14, 855-879. Vaduva, I. (1985). Computer generation of random vectors based on transformation of uniformly distributed vectors, in Proceedings of the Seventh Conference on Probability Theory, ed. M. Iosifescu, pp. 589-598 (NU Science Press, Utrecht). Vajda, I. (1989). Theory of Statistical Inference and Information (Kluwer Academic Publishers, Dordrecht). Vale, C. D. and Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions, Psychometrika 48, 465-471. van Dijk, H. K. (1985). Existence conditions for posterior moments of simultaneous equation model parameters, Report 8551 of the Econometric Institute, Erasmus University, Rotterdam. van Dijk, H. K. (1986). A product of multivariate T densities as upper bound for the posterior kernel of simultaneous equation model parameters. Vijverberg, W. P. M. (1995). Monte Carlo evaluation of multivariate normal probabilities, Journal of Econometrics. Vijverberg, W. P. M. (1996). Monte Carlo evaluation of multivariate Student's t probabilities, Economics Letters 52, 1-6. Vijverberg, W. P. M. (1997). Monte Carlo evaluation of multivariate normal probabilities, Journal of Econometrics 76, 281-307. Vijverberg, W. P. M. (2000). Rectangular and wedge-shaped multivariate normal probabilities, Economics Letters 68, 13-20. Wald, A. (1944). On a statistical problem arising in the classification of an individual into one of two groups, Annals of Mathematical Statistics 15, 145-162.

Wallgren, C. M. (1980). The distribution of the product of two correlated t variates, Journal of the American Statistical Association 75, 996-1000. Walker, G. A. and Saw, J. G. (1978). The distribution of linear combinations oft variables, Journal of the American Statistical Association 73, 876-878.

Wang, O. and Kennedy, W. J. (1990). Comparison of algorithms for bivariate normal probability over a rectangle based on self-validating results from interval analysis, Journal of Statistical Computation and Simulation 37, 13-25. Wang, O. and Kennedy, W. J. (1997). Application of numerical interval analysis to obtain self-validating results for multivariate probabilities in a massively parallel environment, Statistics and Computing 7, 163-171. Watson, G. N. (1958). A Treatise on the Theory of Bessel Functions (Cambridge University Press, Cambridge). Whittaker, E. T. and Watson, G. N. (1952). Modern Analysis (Cambridge University Press, Cambridge). Weir, J. B. de V. (1966). Table of 0 1 percentage points of Behrens's d, Biometrika 53, 267-268. Wooding, R. A. (1956). The multivariate distribution of complex normal variables, Biometrika 43, 212-215. Wu, C. F. J. (1983). On the convergence properties of the EM algorithm, Annals of Statistics 11, 95-103. Wynn, H. P. and Bloomfield, P. (1971). Simultaneous confidence bands for regression analysis (with discussion), Journal of the Royal Statistical Society B 33, 202-217. Yang, Z. Q. and Zhang, C. M. (1997). Dimension reduction and

References

267

Ll-approximation for evaluations of multivariate normal integrals, Chinese Journal of Numerical Mathematics and Applications 19, 82-95. Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics (John Wiley and Sons, New York). Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms, Journal of the American Statistical Association 71, 400-405. Zhuang, X., Huang, Y., Palaniappan, K. and Zhao, Y. (1996). Gaussian density mixture modeling, decomposition and applications, IEEE Transactions on Image Processing 5, 1293-1302. Zografos, K. (1999). On maximum entropy characterization of Pearson's type II and type VII multivariate distributions, Journal of Multivariate Analysis 71, 67-75.

Index

equi, 25, 34, 127, 134, 135, 143, 172, 175, 178, 182, 183, 187

F matrix, 20 F-divergence, 242 L2-divergence, 242

intraclass, 112 quasi-decomposable, 189 singular, 189 cumulant generating function, 121, 122,

t-ratio distribution, 60

Fortran, 153, 154 acceptance-rejection sampling, 157, 158 Appell's hypergeometric function, 32, 33 Appell's polynomials, 10 association, 20

Bartlett's decomposition, 213 Bayesian estimator, 159-161 Bayesian inference, 2, 50, 112, 213, 214, 233-236, 241, 246 Behren-Fisher density, 32, 33 Bessel function first kind, 42 second kind, 42 beta function, 78 Bonferroni's inequalities, 177 BUGS, 226, 227 characteristic function, 36-41, 43-45, 48, 76, 120, 135, 200 Cholesky decomposition, 151, 156 cluster analysis, 52, 212, 244, 245 complementary error function, 143, 144 complete elliptical integral first kind, 48, 78, 79 second kind, 48, 78 concomitant order statistics, 207, 208 cone, 145, 146 confluent hypergeometric function, 47,

205

decision theory, 12 dependence coefficient, 25 digamma function, 22, 23, 204, 218 discriminant analysis, 244 distribution asymmetric multivariate t, 97, 99 bivariate t, 13, 15, 21, 56, 72, 74, 80, 81, 106, 127, 148, 161, 170, 174, 181, 183, 207 bivariate Cauchy, 68, 80 bivariate chi-squared, 71, 73 bivariate normal, 15, 20, 53, 63, 66, 68, 69, 71, 73-75, 80, 119, 129, 148, 154, 170, 174, 209 bivariate Pearson type VII, 80 central bivariate t, 72, 127 central matrix-variate t, 113-116 central multivariate t, 1, 16, 17, 19, 22, 29, 87, 88, 90, 99, 114-117, 126, 131, 135, 140, 165, 170, 172, 175, 182, 204, 205 central univariate F, 20 complex matrix-variate t, 120 complex multivariate t, 119, 120 complex multivariate normal, 119, 120

complex univariate normal, 120 compound normal, 96 conditionally specified bivariate t, 76 Dirichlet, 84, 225 doubly noncentral bivariate t, 71 doubly noncentral multivariate t, 93

71, 90 consistency, 9 convex polyhedra, 148, 150 correlation structure

decomposable, 171, 189, 190 269

270

Index

doubly noncentral univariate t, 71, 93 doubly noncentral univariate chi-squared, 93 Fisher's z, 34 gamma, 226, 239 generalized t, 94, 96, 97, 101 Gumbel, 106 infinite matrix-variate t, 119 inverse gamma, 94 inverted Dirichlet, 126 inverted gamma, 192, 196, 220, 237 matrix-variate t, 97, 112-115, 118, 119

matrix-variate beta, 117 matrix-variate inverse Wishart, 118 matrix-variate normal, 118 mixture of normal, 9 mixture of two normal, 200 multivariate F, 100, 110 multivariate t, 1, 8, 16, 22, 24, 30, 36, 38, 41, 44, 50, 87, 88, 94, 98, 102, 103, 106, 113, 115, 117, 120, 126, 127, 135, 140, 145, 151, 153, 154, 159, 161, 165, 170, 172-174, 183, 184, 187, 188, 191, 204, 210, 213-215, 223, 225-228, 233-236, 239, 241, 243-246

multivariate Bessel, 9 multivariate Cauchy, 1, 9, 22, 25, 40, 41, 100, 192 multivariate elliptical, 2, 94, 96, 100, 120, 121, 123, 125, 126, 196, 200, 201, 240, 243 multivariate logistic, 9

multivariate normal, 1, 2,

9, 12, 13, 15, 18, 24, 31, 50, 52, 73, 87, 90, 93-96, 98-100, 103, 112, 119, 123, 125, 140, 152, 155, 156, 165, 171, 181, 192, 196, 205, 213, 225, 226, 233, 239, 245

multivariate Pearson type II, 9 multivariate Pearson type VII, 2, 9, 40, 41

multivariate uniform, 157, 158, 226 multivariate Wishart, 7 noncentral bivariate t, 63, 66, 68, 69, 73

noncentral complex Wishart, 197 noncentral matrix-variate t, 113 noncentral multivariate t, 1, 23, 87, 90, 93, 139, 140

noncentral multivariate Cauchy, 23 noncentral univariate F, 19, 34 noncentral univariate t, 63 noncentral univariate chi-squared, 71, 209, 239

noncentral Wishart, 197

power exponential, 9 scale mixture of normal, 194, 239 skewed multivariate t, 98, 100, 102-105, 107, 109, 112 skewed multivariate Cauchy, 100 skewed multivariate elliptical, 101 skewed multivariate normal, 98, 103 skewed univariate t, 83, 84, 106, 107, 109

spherical, 9, 96 standard normal, 49, 57, 63, 80, 82, 98, 130, 132, 133, 136, 144, 166, 175, 177, 181-184, 186, 224 stepwise multivariate t, 90, 91 Student's t, 1, 13, 23, 28, 30, 36, 40, 41, 49, 52, 53, 56, 63, 64, 66, 67, 75, 79, 80, 82, 84, 92, 98, 99, 101, 105, 111, 116, 129, 137, 145, 150, 156, 161, 163, 167, 169, 170, 175, 179, 185, 230, 246

symmetric multivariate t, 111 trivariate t, 21, 35, 183 trivariate normal, 67, 121, 123 univariate F, 107, 109, 116, 117, 151, 152, 179, 199, 203, 230, 234, 238 univariate beta, 83, 106, 164 univariate Cauchy, 23, 56, 68 univariate chi-squared, 2, 21, 53, 63, 70, 80, 82, 83, 87, 90, 91, 98, 100, 103, 108, 120, 131, 132, 135, 152, 165, 181, 196, 200-202, 209, 216, 234, 239, 245

univariate Poisson, 93 univariate uniform, 226 Wishart, 193, 213 econometrics, 56, 114 Edgeworth form, 133 EM algorithm, 210-212, 235, 239 entropy, 21-23, 204 entropy loss function, 219, 220 error function, 141 exchangeability, 172 exponential family, 96

factor analysis, 193, 215 fiducial distribution, 45 Fisher's optimal discrimination criterion, 244 forecast error, 13 forecasting, 12, 15 Fortran, 158, 169 Fourier series expansion, 242 Fourier transform, 140

Gauss hypergeometric function, 46, 67, 78, 81, 137

Index Gauss-Hermite quadrature, 134, 143 Gauss-Legendre quadrature, 152 general hypergeometric function, 113 generalized gamma function, 113, 193 Gram-Charlier expansion, 133 Hall's index, 242 Hartley's differential-difference

equation, 133 Hermite polynomial, 132, 136, 176 hierarchical models, 214 Hotelling's T2 statistic, 199, 202, 203, 209, 210

271

missing data imputation, 212 moment generating function, 76, 121-123, 125, 198, 205

monotone data augmentation, 213 Monte Carlo algorithm, 152, 157 Monte Carlo simulation, 161, 162 multiple comparison procedure, 154, 158 multiple correlation coefficient, 91 multiple decision problem, 245 multiple regression analysis, 91 multiple stationary normal time series, 120

multivariate t model, 191-194, 219, 221, 222

importance sampling, 161, 163, 164 imputation multiple, 212, 213 single, 212

incomplete beta function, 62, 128, 139,

generalized, 196, 222 multivariate t statistic, 198, 199 multivariate skewness, 102 multivariate tail-dependence, 243 mutual information, 24, 25

148, 149

incomplete beta function ratio, 139, 232 incomplete gamma function ratio, 47 infinitely divisible, 41, 43 information matrix, 217-219, 235 inversion theorem, 135

nonlinear model, 239, 240 nonlinear regression, 235, 239 multivariate, 235 normal law, 198, 199

Kullback-Leibler number, 24, 204, 205 kurtosis, 28, 121-123, 125, 126, 200, 223, 224

online environmental learning, 246 order statistics, 186, 207, 208 orthant symmetry, 202 orthogonal expansion index, 242 orthogonally invariant, 97, 117

Levy representation, 41, 43 Laguerre polynomial, 177 Laplacian T-approximation, 214 Laplacian approximation, 214 lattice rule algorithm, 157, 158 least absolute deviation estimator, 243 linear inequalities, 158 linear model Bayesian, 102, 233 classical, 228, 239 general, 237 indexed, 235 linear regression, 213 multivariate, 213 linear simultaneous equation model, 114 load, 158

parabolic cylinder function, 90, 144 partial regression coefficient, 216 percentage points, 57, 59, 174-178, 180-185, 187-189 Poisson mixture, 93 portfolio optimization, 243 predictive distribution, 114, 232, 233 prior diffuse, 50, 52, 114, 234 improper, 240 natural conjugate, 52, 114, 234 proper, 240 probability inequalities, 165, 169-172 projection index, 241, 242 projection pursuit, 241

logit, 164

MacDonald function, 36, 38, 40, 42 macroeconomic modeling, 114 Mahalanobis distance, 210

quadrant dependence, 20 quadratic form, 19, 20, 34, 50, 96, 100,

MANOVA, 239

104, 115, 125, 197, 198, 218 quadratic loss function, 194, 222 quadrature formulas, 137

maximum entropy characterization, 23 distribution, 23 maximum probability content, 160 microeconomic modeling, 114

Renyi distances, 28 Renyi information, 26-28 randomized block design, 178 ranking and selection, 56

Index

272 rejection algorithm, 226 reliability, 152 robust estimation, 235 Schur-concave, 2, 172 score function, 215, 217 score test, 215, 216 security returns, 246 Shannon entropy, 27 Simpson's rule, 169 skewness, 102, 121, 217, 223, 224 slope coefficient, 243 spectral decomposition, 195, 221

spectral density matrix, 120 speech recognition, 246 squared error loss function, 221 standardized cumulant, 133 StatLib, 212 Stein loss function, 219 stock market problems, 192 strength, 158 Student's t index, 242 Studentized maximum and minimum modulus tests, 139 Studentized statistics, 134 Taylor series expansion, 35, 230 multivariate, 154 triangular decomposition, 195 trigamma function, 235 Tukey's procedure, 154

variance component model, 214

Wishart matrix, 191-193, 196, 204, 219, 221, 222

Multivariate T-Distributions and Their Applications

Multivariate t distributions and their applications

Multivariate T-distributions and their applications

Multivariate Approximation and Applications

Multivariate statistical inference and applications

Multivariate statistical inference and applications

Orders and their Applications

Metagraphs and Their Applications

Wavelets and their applications

Metagraphs and Their Applications

Wavelets and their applications

Cellulases and Their Applications

Metaclasses and Their Applications

Geosynthetics and their applications

Metaclasses and Their Applications

Wavelets and their Applications

Wavelets and Their Applications

Semigroups and their applications

Continuous Multivariate Distributions. Models and applications

Multilevel Modeling: Methodological Advances, Issues, and Applications (Multivariate Applications Series)

Baecklund transformations and their applications

Integral equations and their applications

Maximum principles and their applications

Bioceramics and Their Clinical Applications

Wavelet Transforms and Their Applications

Exponential Sums and Their Applications

Latin Squares and Their Applications

Spectral methods and their applications

Latin Squares and Their Applications

Integral Transforms and Their Applications

Orthogonal Polynomials and their Applications

Multivariate T-Distributions and Their Applications