Table Of ContentTheory of Multivariate
Statistics
Martin Bilodeau
David Brenner
Springer
A la m´emoire de mon p`ere, Arthur, a` ma m`ere, Annette, et `a Kahina.
M. Bilodeau
To Rebecca and Deena.
D. Brenner
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Preface
Our object in writing this book is to present the main results of the mod-
ern theory of multivariate statistics to an audience of advanced students
who would appreciate a concise and mathematically rigorous treatment of
that material. It is intended for use as a textbook by students taking a
first graduate course in the subject, as well as for the general reference of
interestedresearchworkerswhowillfind,inareadableform,developments
from recently published work on certain broad topics not otherwise easily
accessible,as,forinstance,robustinference(usingadjustedlikelihoodratio
tests)andtheuseofthebootstrapinamultivariatesetting.Thereferences
contains over 150 entries post-1982. The main development of the text is
supplemented by over 135 problems, most of which are original with the
authors.
A minimum background expected of the reader would include at least
two courses in mathematical statistics, and certainly some exposure to the
calculusofseveralvariablestogetherwiththedescriptivegeometryoflinear
algebra.Ourbookis,nevertheless,inmostrespectsentirelyself-contained,
althoughadefiniteneedforgenuinefluencyingeneralmathematicsshould
not be underestimated. The pace is brisk and demanding, requiring an in-
tense level of active participation in every discussion. The emphasis is on
rigorousproofandderivation.Theinterestedreaderwouldprofitgreatly,of
course,frompreviousexposuretoawidevarietyofstatisticallymotivating
material as well, and a solid background in statistics at the undergraduate
level would obviously contribute enormously to a general sense of famil-
iarity and provide some extra degree of comfort in dealing with the kinds
of challenges and difficulties to be faced in the relatively advanced work
viii Preface
of the sort with which our book deals. In this connection, a specific intro-
duction offering comprehensive overviews of the fundamental multivariate
structures and techniques would be well advised. The textbook A First
Course in Multivariate Statistics by Flury (1997), published by Springer-
Verlag, provides such background insight and general description without
getting much involved in the “nasty” details of analysis and construction.
This would constitute an excellent supplementary source. Our book is in
most ways thoroughly orthodox, but in several ways novel and unique.
In Chapter 1 we offer a brief account of the prerequisite linear algebra
asitwillbeappliedinthesubsequentdevelopment.Someofthetreatment
is peculiar to the usages of multivariate statistics and to this extent may
seem unfamiliar.
Chapter 2 presents in review, the requisite concepts, structures, and
devices from probability theory that will be used in the sequel. The ap-
proachtakeninthefollowingchaptersrestsheavilyontheassumptionthat
this basic material is well understood, particularly that which deals with
equality-in-distribution and the Cram´er-Wold theorem, to be used with
unprecedented vigor in the derivation of the main distributional results in
Chapters 4 through 8. In this way, our approach to multivariate theory
is much more structural and directly algebraic than is perhaps traditional,
tiedinthisfashionmuchmoreimmediatelytothewayinwhichthevarious
distributions arise either in nature or may be generated in simulation. We
hopethatreaderswillfindtheapproachrefreshing,andperhapsevenabit
liberating, particularly those saturated in a lifetime of matrix derivatives
and jacobians.
As a textbook, the first eight chapters should provide a more than ade-
quate amount of material for coverage in one semester (13 weeks). These
eight chapters, proceeding from a thorough discussion of the normal dis-
tribution and multivariate sampling in general, deal in random matrices,
Wishart’s distribution, and Hotelling’s T2, to culminate in the standard
theory of estimation and the testing of means and variances.
Theremainingsixchapterstreatofmorespecializedtopicsthanitmight
perhaps be wise to attempt in a simple introduction, but would easily be
accessible to those already versed in the basics. With such an audience in
mind, we have included detailed chapters on multivariate regression, prin-
cipal components, and canonical correlations, each of which should be of
interesttoanyonepursuingfurtherstudy.Thelastthreechapters,dealing,
inturn,withasymptoticexpansion,robustness,andthebootstrap,discuss
conceptsthatareofcurrentinterestforactiveresearchandtakethereader
(gently) into territory not altogether perfectly charted. This should serve
to draw one (gracefully) into the literature.
Theauthorswouldliketoexpresstheirmostheartfeltthankstoeveryone
who has helped with feedback, criticism, comment, and discussion in the
preparation of this manuscript. The first author would like especially to
convey his deepest respect and gratitude to his teachers, Muni Srivastava
Preface ix
oftheUniversityofTorontoandTakeakiKariyaofHitotsubashiUniversity,
whogavetheirunstintingsupportandencouragementduringandafterhis
graduate studies. The second author is very grateful for many discussions
with Philip McDunnough of the University of Toronto. We are indebted
to Nariaki Sugiura for his kind help concerning the application of Sug-
iura’s Lemma and to Rudy Beran for insightful comments, which helped
to improve the presentation. Eric Marchand pointed out some errors in
the literature about the asymptotic moments in Section 8.4.1. We would
like to thank the graduate students at McGill University and Universit´e
de Montr´eal, Gulhan Alpargu, Diego Clonda, Isabelle Marchand, Philippe
St-Jean, Gueye N’deye Rokhaya, Thomas Tolnai and Hassan Younes, who
helpedimprovethepresentationbytheircarefulreadingandproblemsolv-
ing. Special thanks go to Pierre Duchesne who, as part of his Master
Memoir, wrote and tested the S-Plus function for the calculation of the
robust S estimate in Appendix C.
M. Bilodeau
D. Brenner
