Bilinear Forms and Zonal Polynomials

384 Pages·1995·11.238 MB·English

by  A. M. Mathai, Serge B. Provost, Takesi Hayakawa (auth.)

Lecture Notes in Statistics 102
Edited by P. Diggle, S. Fienberg, K. Krickeberg,
1. Olkin, N. Wennuth
A.M. Mathai, Serge B. Provost,
and Takesi Hayakawa
Bilinear Forms and Zonal
Polynomials
Springer-Verlag
New York. Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
A.M. Mathai Serge B. Provost
Department of Mathematics Department of Statistical and
McGill University Actuarial Sciences
Montreal University of Western Ontario
Quebec, Canada H3A 2K6 London
Ontario, Canada N6A 5B6
Takesi Hayakawa
Department of Economics
Hitotubashi University
2-1 Naka, Kunitachi
Tokyo 186
Japan
library of Congress Cataloging-in-Publication Data Available
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© 1995 Springer-Verlag New Yorle, Inc.
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987654321
ISBI\-13: 978-0-387-94522-4 e-ISBN-13: 978-1-4612-4242-0
DOT: 10.1007/978-1-4612-4242-0
To my grandparents and my godmother Gisele Bedard
S.B.P.
To my wife Yuhko
T.H.
Preface
The book deals with bilinear forms in real random vectors and their generalizations
as well as zonal polynomials and their applications in handling generalized quadratic
and bilinear forms. The book is mostly self-contained. It starts from basic principles
and brings the readers to the current research level in these areas. It is developed with
detailed proofs and illustrative examples for easy readability and self-study. Several
exercises are proposed at the end of the chapters. The complicated topic of zonal
polynomials is explained in detail in this book. The book concentrates on the theoretical
developments in all the topics covered. Some applications are pointed out but no detailed
application to any particular field is attempted.
This book can be used as a textbook for a one-semester graduate course on quadratic
and bilinear forms and/or on zonal polynomials. It is hoped that this book will be a
valuable reference source for graduate students and research workers in the areas of
mathematical statistics, quadratic and bilinear forms and their generalizations, zonal
polynomials, invariant polynomials and related topics, and will benefit statisticians,
mathematicians and other theoretical and applied scientists who use any of the above
topics in their areas.
Chapter 1 gives the preliminaries needed in later chapters, including some Jacobians
of matrix transformations. Chapter 2 is devoted to bilinear forms in Gaussian real ran
dom vectors, their properties, and techniques specially developed to deal with bilinear
forms where the standard methods for handling quadratic forms become complicated.
Distributional aspects and Laplacianness, a concept analogous to chi-squaredness for
quadratic forms, are examined in detail.
Various distributional results on quadratic forms in elliptically contoured and spher
ically symmetric vectors are presented in Chapter 3. The central and noncentral distri
butions, the moments, Cochran's theorem as well as quadratic forms of random idem
potent matrices are discussed, and the robust properties of certain test statistics are
studied. Several results also apply to bilinear forms.
Chapter 4 proposes a systematic development of the theory of zonal polynomials,
including worked examples and related topics. Many of these results are used in Chap
ter 5 which deals with the distribution of generalized bilinear and quadratic forms. The
theoretical results and applications are brought up to the current research level. In
variant polynomials, which are an extension of zonal polynomials, are discussed in an
appendix. Some fundamental results and useful coefficients relating to these polynomi
als are tabulated.
vii
Vlll PREFACE
We are very grateful to Professor David R. Bellhouse, Chairman of the Department
of Statistical and Actuarial Sciences at the University of Western Ontario, for making
departmental personnel and equipment available to us. We would like to thank Dr
Edmund Rudiuk and John Kawczak for their comments and suggestions on various parts
of the book. We also wish to express our appreciation to Alicia Pleasence, Motoko Yuasa
and Yoriko Fukushima who patiently and expertly typed most of Chapters 3, 4 and 5,
including the Appendix. The financial support of the Natural Sciences and Engineering
Research Council of Canada is gratefully acknowledged.
1 February 1995 Arak M. Mathai
Serge B. Provost
Takesi Hayakawa
Contents
PREFACE vii
Chapter 1
PRELIMINARIES
1.0 Introduction . 1
1.1 Jacobians of Matrix 'lransfonnations . . . . . . . . . 1
1.1& Some Frequently Used Jacobians in the Real Case 2
1.2 Singular and Nonsingular Nonnal Distributions 6
1.2& Normal Distribution in the Real Case 6
1.2b The Moment Generating FUnction
for the Real Normal Distribution. . 7
1.3 Quadratic Forms in Normal Variables. . . 8
1.3& Representations of a Quadratic Fonn 8
1.3b Representations of the m.g.f. of a Quadratic Expression 9
1.4 Matrix-variate Gamma and Beta FUnctions . . . 10
1.4a Matrix-variate Gamma, Real Case . . . . . . . . . 10
1.4b Matrix-variate Gamma Density, Real Case ..... 12
lAc The m.g.f. of a Matrix-variate Real Gamma Variable 13
1.4d Matrix-variate Beta, Real Case 13
1.5 Hypergeometric Series, Real Case ........... . 16
Chapter 2
QUADRATIC AND BILINEAR FORMS IN NORMAL VECTORS
2.0 Introduction. . . . . . . . . 17
2.1 Various Representations. . . . 19
2.2 Density of a Gamma Difference 20
2.2a Some Particular Cases . 22
2.3 Noncentral Gamma Difference . 24
2.4 Moments and Cumulants of Bilinear Fonns 27
2.4a Joint Moments and Cumulants
of Quadratic and Bilinear Forms . . 28
ix
x CONTENTS
2.4a.l The First Few Joint Cumulants of
a Bilinear Form and a Quadratic Form
in the Nonsingular Normal Case . 31
2.4a.2 The First Few Cumulants of a
Quadratic Form in the Nonsingular Normal Case 33
2.4a.3 The First Few Cumulants of a
Bilinear Form in the Nonsingular Normal Case 34
2.4b Joint Cumulants of Bilinear Forms . 35
2.4b.l Tbe First Few Joint Cumulants of
of Two Bilinear Forms 37
2.4c Moments and Cumulants in tbe Singular Normal Case 39
2.4c.l Tbe First Few Cumulants of
a Bilinear Form in tbe Singular
and Nonsingular Normal Cases 40
2.4d Cumulants of Bilinear Expressions 42
2.4d.l Some Special Cases of Bilinear Expressions 42
2.5 Laplacianness of Bilinear Forms 45
2.5a Quadratic and Bilinear Forms
in tbe Nonsingular Normal Case 47
2.5b NS Conditions for tbe Noncorrelated Normal Case 50
2.00 Quadratic and Bilinear Forms
in tbe Singular Normal Case 52
2.5d Noncorrelated Singular Normal Case 54
2.5e Tbe NS Conditions for a Quadratic Form to be NGL 55
2.6 Generalizations to Bilinear and Quadratic Expressions 57
2.6a Bilinear and Quadratic Expressions
in tbe Nonsingular Normal Case . 57
2.6b Bilinear and Quadratic Expressions
in tbe Singular Normal Case . 61
2.7 Independence of Bilinear and Quadratic Expressions 63
2.7a Independence of a Bilinear
and a Quadratic Form 63
2.7h Independence of Two Bilinear Forms . 65
2.7c Independence of Quadratic Expressions:
Nonsingular Normal Case 67
2.7d Independence in tbe Singular Normal Case 70
2.8 Bilinear Forms and Noncentral Gamma Differences 74
2.8a Bilinear Forms in tbe Equicorrelated Case . 75
2.8b Noncentral Case . 78
2.9 Rectangular Matrices . 80
2.9a Matrix-variate Laplacian 82
2.9b Tbe Density of 521 83
2.9c A Particular Case 84
Exercises 86
CONTENTS xi
Chapter 3
QUADRATIC AND BILINEAR FORMS
IN ELLIPTICALLY CONTOURED DISTRIBUTIONS
3.0 Introduction 89
3.1 Definitions and Basic Results . . . 91
3.2 Moments of Quadratic Forms . . . 104
3.3 The Distribution of Quadratic Forms 111
3.4 Noncentral Distribution ..... 116
3.5 Quadratic Forms in Random Matrices 122
3.6 Quadratic Forms of Random Idempotent Matrices 123
3.7 Cochran's Theorem ............ . 131
3.8 Test Statistics for Elliptically Contoured Distributions 138
Sample Correlation Coefficient 139
Likelihood Ratio Criteria 145
Exercises 155
Chapter 4
ZONAL POLYNOMIALS
4.0 Introduction 163
4.1 Wishart Distribution 163
4.2 Symmetric Polynomials 166
4.3 Zonal Polynomials . . 171
4.4 Laplace 1Tansform and Hypergeometric .FUnction 193
4.5 Binomial Coefficients . 210
4.6 Some Special .FUnctions 215
Exercises .. 234
Table 4.3.2(a) 243
Table 4.3.2(b) 244
Table 4.4.1 246
Chapter 5
GENERALIZED QUADRATIC FORMS
5.0 Introduction 247
5.1 A Representation of the Distribution of
a Generalized Quadratic Form . 248
5.2 An Alternate Representation 255
5.3 The Distribution of the Latent
Roots of a Quadratic Form . . 258
5.4 Distributions of Some .FUnctions of X AX' 264
5.5 Generalized Hotelling's T~ ..... . 268
5.6 Anderson's Linear Discriminant FUnction 279
5.7 Multivariate Calibration ...... . 286
5.8 Asymptotic Expansions of the Distribution of a Quadratic Form 294
Exercises ....................... . 300
xii CONTENTS
Table 5.6.1 317
Table 5.6.2 318
Appendix
INVARIANT POLYNOMIALS
Appendix A.1 Representation of a Group ..... . 319
Appendix A.2 Integration of the Representation Matrix
over the Orthogonal Group ..... 323
Appendix A.3 Fundamental Properties of C;'>'(X, Y) 325
Bibliography . 353
Glossary of Symbols . 363
Author Index . 371
Subject Index 375