Table Of ContentElliptically Contoured Models in Statistics
Mathematics and Its Applications
ManagingEditor:
M.HAZEWINKEL
CentreforMathematicsandComputerScience,Amsterdam,TheNetherlands
Volume240
Elliptically Contoured
Models in Statistics
by
A. K. Gupta
Department of Malhematics and Stal;stics,
Bowling Green Slale University,
Bowling Green, OhiQ. U.sA.
and
T. Varga
AB-AEGON Generallnsurance Company,
Budapesl, Hungary
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Dala
Gupta. A. K. (ArJun K.l, 1938-
E111ptlcal1v contoured .odals In stathtlcs I bV A.K. Gupta and T.
Varga.
p. C• . -- (Mathe.nlcs and 111 appllcatlons ; v. 2401
Includes blbl100raphlcal referenclS and Indens.
ISBN 978-94-010-4719-7 ISBN 978-94-011-1646-6 (eBook)
DOI 10.1007/978-94-011-1646-6
1. DIstrIbution IProbabll1tv theorv) 2. Multlvarlate analvsls.
I. Varga, T. (T .. esl, 1960- Ir. Tltle. Ill. Sl!rlu,
Matheut'c$ and ItS appllcatlOns (Kluwer Acadu'c Publ1shersl ; v.
240.
0A273.S.G87 1993
619.2'4--dc20 92-44826
ISBN 918-94-010-4119-1
Printed on acid-free poper
AU Rights Reserved
e
1993 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1993
Softcover reprint ofthe hardcover Ist edition 1993
as specified on appropriate pages within.
No pan ofthe material protected by this copyright notice may be reproduced or
utilized in any fonn or by any means, electronic or mechanical,
including photocopying, recording or by any infonnalion slOrage and
relrieval syslem, withoul wrillen permission from the copyrighl owner,
Dedicated to the memory of my mother and father.
AKG
TABLE OF CONTENTS
PREFACE ix
CHAPTER 1. PRELIMINARIES 1
1.1 Introduction and Literature Review 1
1.2 Notations 2
1.3 Some Results from Matrix Algebra 6
1.4 A Functional Equation 17
CHAPTER 2. BASIC PROPERTIES 19
2.1 Definition 19
2.2 Probability Density Function 26
2.3 Marginal Distributions 31
2.4 Expected Value and Covariance 32
2.5 Stochastic Representation 35
2.6 Conditional Distributions 47
2.7 Examples 70
CHAPTER 3. PROBABILITY DENSITY FUNCfION AND
EXPECTEDVALUES 80
3.1 Probability Density Function 80
3.2 More on Expected Values 91
CHAPTER 4. MIXTURE OF NORMAL DISTRIBUTIONS 130
4.1 Mixture by Distribution Functions 130
4.2 Mixture by Weighting Functions 150
vii
viii TABLE OF CONTENTS
CHAPTER 5. QUADRATIC FORMS AND OTHER FUNCTIONS OF
ELLIPTICALLY CONTOURED MATRICES 162
5.1 Cochran's Theorem 162
5.2 Rank of Quadratic Forms 179
5.3 Distribution of Invariant Matrix Variate Functions 182
CHAPTER 6. CHARACTERIZATION RESULTS 190
6:1 Characterizations Based on Invariance 190
6.2 Characterizations of Normality 193
CHAPTER 7. ESTIMATION 224
7.1 Maximum Likelihood Estimators of the Parameters 224
7.2 Properties of the Estimators 238
CHAPTER 8. HYPOTHESIS TESTING 252
8.1 General Results 252
8.2 Two Models 258
8.3 Testing Criteria 263
CHAPTER 9. LINEAR MODELS 285
9.1 Estimation of the Parameters in the Multivariate
Linear Regression Model 285
9.2 Hypothesis Testing in the Multivariate Linear
Regression Model 297
9.3 Inference in the Random Effects Model 302
REFERENCES 311
AUTHOR INDEX 321
SUBJECT INDEX 323
PREFACE
In multivariate statistical analysis, elliptical distributions have
recently provided an alternative to the normal model. Most of the work,
however, is spread out in journals throughout the world and is not easily
accessible to the investigators. Fang, Kotz, and Ng presented a systematic
study of multivariate elliptical distributions, however, they did not discuss
the matrix variate case. Recently Fang and Zhang have summarized the
results of generalized multivariate analysis which include vector as well as
the matrix variate distributions. On the other hand, Fang and Anderson
collected research papers on matrix variate elliptical distributions, many of
them published for the first time in English. They published very rich
material on the topic, but the results are given in paper form which does not
provide a unified treatment of the theory. Therefore, it seemed appropriate
to collect the most important results on the theory of matrix variate
elliptically contoured distributions available in the literature and organize
them in a unified manner that can serve as an introduction to the subject.
The book will be useful for researchers, teachers, and graduate
students in statistics and related fields whose interests involve multivariate
statistical analysis. Parts ofthis book werepresentedby Arjun K Gupta as a
one semester course at Bowling Green State University. Some new results
have also been included which generalize the results in Fang and Zhang.
Knowledge ofmatrix algebra and statistics at the level ofAnderson is
assumed. However, Chapter 1summarizes some results of matrix algebra.
This chapter also contains a brief review of the literature and a list of
mathematical symbols used in the book.
Chapter 2 gives the basic properties of the matrix variate elliptically
contoured distributions, such as the probability density function and
expected values. It also presents one of the most important tools of the
theory of elliptical distributions, the stochastic representation.
ix
x PREFACE
The probability density function and expected values are investigated
in detail in Chapter 3.
Chapter 4 focuses on elliptically contoured distributions that can be
represented as mixtures of normal distributions.
The distributions of functions of random matrices with elliptically
contoured distributions are discussed in Chapter 5. Special attention is given
to quadratic forms.
Characterization results are given in Chapter 6.
The last three chapters are devoted to statistical inference. Chapter 7
focuses on estimation results, whereas Chapter 8 is concerned with
hypothesis testing problems. Inference for linear models is studied in
Chapter 9. Finally, an up to date bibliography has been provided, along with
author and subject indexes.
We would like to thank the Department of Mathematics and
Statistics, Bowling Green State University, for supporting our endeavour
and for providing the necessary facilities to accomplish the task. The first
author is thankful to the Biostatistics Department, University of Michigan,
for providing him the opportunity to organize the material in its final form.
Thanks are also due to Professors A. M. Kshirsagar, D. K. Nagar, M. Siotani,
and J. Tang for manyhelpful discussions. He would also like to
acknowledge his wife, Meera, and his children, Alka, Mita, and Nisha for
their support throughout the writing of the book.
Particular gratitude is rendered to Cynthia Patterson for her efficient
technical word processing. We also express our appreciation to the publisher
and especially to Dr. D. J. Lamer for his sustained interest in the project.
September 1992 A. K. Gupta
T. Varga
CHAPTER 1
PRELIMINARIES
1.1. INTRODUCTION AND LITERATURE REVIEW
Matrix variate distributions have been studied by statisticians for a
long time. The first results on this topic were published by Hsu and Wishart.
These distributions proved to be useful in statistical inference. For example,
the Wishart distribution is essential when studying the sample covariance
matrix in the multivariate normal theory. Random matrices can also be
used to describe repeated measurements on multivariate variables. In this
case, the assumption of the independence of the observations, a commonly
used condition in statistical analysis, is often not feasible. When analyzing
data sets like these, the matrix variate elliptically contoured distributions can
beused to describe the dependence structure ofthe data. This is a rich class
of distributions containing the matrix variate normal, contaminated normal,
Cauchy and Student's t-distributions. The fact that the distributions in this
class possess certain properties, similar to those of the normal distribution,
makes them especially useful. For example, many testing procedures
developed for the normal theory to test various hypotheses can be used for
this class of distributions, too.
Matrix variate elliptically contoured distributions represent an
extension of the concept of elliptical distributions from the vector to the
matrix case. Important distribution results on vector variate elliptical
distributions were derived by Kelker (1970),Chu (1973), Dawid (1977) and
Cambanis, Huang, and Simons (1981). Quadratic forms in elliptical
distributions werestudied by Cacoullos and Koutras (1984), Fang and Wu
(1984), Anderson and Fang (1987), and Smith (1989). Problems related to
moments were considered by Berkane and Bentler (1986a). Characterization
1
