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    Aspects of Multivariate Statistical Theory
    Aspects of Multivariate Statistical Theory ROBB J. MUIRHEAD Senior Statistical Scientist PJizer Global Research and Development New London, Conneclicut @E-+!&CIENCE A JOHN WILEY & SONS, INC., PUBLICATION
    Copyright 0 1982,2005 hy John Wiley & Sotis, Inc. All rights rcscrvcd Published by John Wiley & Sons, Inc., Hohokcn, Ncw Icrscy Published simultaneously in Canada. No part of this publication may he reproduccd, storcd in a rctricval systeni or transmittcd in any lorm or by any mcans, electronic, niechanical. photncopying. rm)rding. scuniiing. or othcrwisc, cxccpt as pcrmitted under Section 107 o r 108 of the 1976 United Statcs (‘opyright Act. withnut cithcr the prior written permission of the Puhlishcr. or autlioiiiation through paynicnt of thc appropriate p c r a p y fec to the Copyright Clcarance Center, Inc., 222 Iisclaimer of Warranty: While the puhlishcr and author havc uscd thcir hcst efforts in preparing this bcwk, they makc no rcprescntations or warrantics with respwt to the accuracy or completeness of‘the contents ofthis hwk and spcwifically disclaim any iiiiplicd warranties of merchantability or fitness for a particular purpose No warranty may hc crcatcd or extended hy sales represcntatives or written snlcs iiiatcrials. The advicc illid stratcgicr contained herein may not be suitahle I‘or your situation. You should conslilt with’a profisioital where appropriate. Neither the puhlishcr nor author shall bc liuhlc Tor ally loss ol’prolit or any otlicr commercial damages, including hut not limited tc;rpccial. incidcntal. coitxequciitial. or otlicr damages. For general information on our other products and services or fnr technical support, plcasc contact our Customer Care Department within thc 1J.S. at (800) 762-2974, outsidc the U.S. at (3 17) 572- 3993 or fax (3 17) 572-4002. Wiley also publishes its hooks in a variety nl’electroitic hrtnats. Some cottictit Iliac appcars i t i piint may not be available in electronic format. For inlimnation ahout Wiley products, visit our web site at www.wiley.com. Library of Congress Catalogin~-in-PubNtion ih uvuiluhle. ISBN- I 3 978-0-47 1-76985-9 ISBN-I0 0-471-76985-1 Printed in the United States of Amcrica 1 0 9 8 7 6 5 4 3 2 I
    To Nan and Bob arid Maria and Mick
    Preface This book has grown out of lectures given in first- and second-year graduate courses at Yale University and the University of Michigan. It is designed as a text for graduate level courses in multivariate statistical analysis, and I hope that it may also prove to be useful as a reference book for research workers interested in this area. Any person writing a book in multivariate analysis owes a great debt to T. W. Anderson for his 1958 text, An Introduction 10 Multivariate Statistical Analysis, which has become a classic in the field. This book synthesized various subareas for the first time in a broad overview of the subject and has influenced the direction of recent and current research in theoretical multi- variate analysis. It is also largely responsible for the popularity of many of the multivariate techniques and procedures in common use today. The current work builds on the foundation laid by Anderson in 1958 and in large part is intended to describe some of the developments that have taken place since then. One of the major developments has been the introduction of zonal polynomials and hypergeometric functions of matrix argument by A. T. James and A. G. Constantine. To a very large extent these have made possible a unified study of the noncentral distributions that arise in multivariate analysis under the standard assumptions of normal sampling. This work is intended to provide an introduction to some of this theory. Most books of this nature reflect the author’s tastes and interests, and this is no exception. The main focus of this work is on distribution theory, both exact and asymptotic. Multivariate techniques depend heavily on latent roots of random matrices; all of the important latent root distribu- tions are introduced and approximations to them are discussed. In testing problems the primary emphasis here is on likelihood ratio tests and the distributions of likelihood ratio test statistics, The noncentral distributions vii
    Prejulte viii are needed to evaluate power functions. Of course, in the absence of “best” tests simply computing power functions is of little interest; what is needed is a comparison of powers of competing tests over a wide range of alternatives. Wherever possible the results of such power studies in the literature are discussed. I I should be mentioned, however, that although the emphasis is on likelihood ratio statistics, many of the techniques introduced here for studying and approximating their distributions can be applied to other test statistics as well. A few words should be said about the material covered in the text. Matrix theory is used extensively, and matrix factorizations are extremely important. Most of the relevant material is reviewed in the Appcndix, but some results also appear in the text and as exercises. Chapter I introduces the multivariate normal distribution and studies its properties, and also provides an introduction to spherical and elliptical distributions. These form an important class of non-normal distributions which have found increasing use in robustness studies where the aim is to determine how sensitive existing multivariate techniques are to multivariate normality assumptions. In Chapter 2 many of the Jacobians of transformations used in the text are derived, aiid a brief introduction to invariant measures via exterior differen- tial forms is given. A review of rnatrix Kronecker or direct products is also included here, The reason this is given at this point rather than in the Appendix is that very few of the students that I have had in multivariate analysis courses have been familiar with this product, which is widely used in later work. Chapter 3 deals with the Wishart and multivariate beta distributions and their properties. Chapter 4, on decision-theoretic estima- tion of the parameters of a multivariate normal distribution, is rather an anomaly. I would have preferred to incorporate this topic in one of the other chapters, but there seemed to be no natural place for it. The niaterial here is intended only as an introduction and certainly not as a review of the current state of the art. Indeed, only admissibility (or rather, inadmissibility) results are presented, and no mention is even made of Bayes procedures. Chapter 5 deals with ordinary, multiple, and partial correlation coefficients. An introduction to invariance theory and invariant tests is given in Chapter 6. It may be wondered why this topic is included here in view of the coverage of the relevant basic material in the books by E. L.. L.ehmann, Testing Statistical Hypotheses, and T. S . Ferguson, Mathenintical Statistics: A Decision Theoretic Approach. The answer is that most of the students that have taken my multivariate analysis courses have been unfamiliar with invariance arguments, although they usually meet them in subsequent courses. For this reason I have long felt that an introduction to invariant tests in a multivariate text would certainly not be out of place.
    Preluce ix Chapter 7 is where this book departs most significantly from others on multivariate statistical theory. Here the groundwork is laid for studying the noncentral distribution theory needed in subsequent chapters, where the emphasis is on testing problems in standard multivariate procedures. Zonal polynomials and hypergeometric functions of matrix argument are intro- duced, and many of their properties needed in later work are derived. Chapter 8 examines properties, and central and noncentral distributions, of likelihood ratio statistics used for testing standard hypotheses about covari- ance matrices and mean vectors. An attempt is also made here to explain what happens if these tests are used and the underlying distribution is non-normal. Chapter 9 deals with the procedure known as principal compo- nents, where much attention is focused on the latent roots of the sample covariance matrix. Asymptotic distributions of these roots are obtained and are used in various inference problems. Chapter 10 studies the multivariate general linear model and the distribution of latent roots and functions of them used for testing the general linear hypothesis. An introduction to discriminant analysis is also included here, although the coverage is rather brief. Finally, Chapter I I deals with the problem of testing independence between a number of sets of variables and also with canonical correlation analysis. The choice of the material covered is, of course, extremely subjective and limited by space requirements. There are areas that have not been men- tioned and not everyone will agree with my choices; I do believe, however, that the topics included form the core of a reasonable course in classical multivariate analysis. Areas which are not covered in the text include factor analysis, multiple time series, multidimensional scaling, clustering, and discrete multivariate analysis. These topics have grown so large that there are now separate books devoted to each. The coverage of classification and discriminant analysis also is not very extensive, and no mention is made of Bayesian approaches; these topics have been treated in depth by Anderson and by Kshirsagar, Multivariate Analysis, and Srivastava and Khatri, An Introduction to Multioariate Statistics, and a person using the current work as a text may wish to supplement it with material from these references. This book has been planned as a text for a two-semester course in multivariate statistical analysis. By an appropriate choice of topics it can also be used in a one-semester course. One possibility is to cover Chapters 1, 2, 3, 5, and possibly 6, and those sections of Chapters 8, 9, 10 and 1 I which do not involve noncentral distributions and consequently do not utilize the theory developed in Chapter 7. The book is designed so that for the most part these sections can be easily identified and omitted if desired. Exercises are provided at the end of each chapter. Many of these deal with points
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