《多元统计分析》课程教学资源（书籍文献，多元分析）Robb J. Muirhead（2005）Aspects of Multivariate Statistical Theory, Wiley.pdf_大学文库

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viiPrefaceare needed to evaluate power functions. Of course, in the absence of "best"tests simply computing power functions js of little interest; what is nceded isa comparison of powers of competing tests overa wide range of alternatives.Wherever possible the results of such power studies in the literature arediscussed. It should be mentioned, however, that although the emphasis ison likelihood ratio statistics, many of the techniques introduced here forstudying and approximating their distributions can be applicd to other teststatisticsas well.A few words should be said about the material covered in the text.Matrix theory is used extensively, and matrix factorizations are extremelyimportant.Most of the relevant material is reviewed in the Appendix, butsome results also appear in the text and as exercises. Chapter I introducesthe multivariate normal distribution and studies its properties, and alsoprovides an introduction to spherical and elliptical distributions.These forman important class of non-normal distributions which have found increasinguse in robustness studies where the aim is to determine how sensitiveexisting multivariate techniques are to multivariate normality assumptions.InChapter2many of the Jacobians of transformations used in the text arederived, and a brief introduction to invariant measures via exterior differen-tial forms is given. A review of matrix Kronecker or direct products is alsoincluded here. The reason this is given at this point rather than in theAppendix is that very few of the students that I have had in multivariateanalysis courses have been familiar with this product, which is widely usedin later work. Chapter 3 deals with the Wishart and multivariate betadistributions and their properties. Chapter 4, on decision-theoretic estima-tion of the parameters of a multivariate normal distribution, is rather ananomaly. I would have preferred to incorporate this topic in one of theother chapters, but there seemed to be no natural place for it. The materialhere is intended only as an introduction and certainly not as a review of thecurrent 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 Chapter6. It may be wondered why this topic is included here in view of thecoverage of the relevant basic material in the books by E. L.. Lehmann,Testing Statistical Hypotheses, and T.S.Ferguson, Mathematical Statistics:A DecisionTheoretic Approach.The answer is that most of the students thathave taken my multivariate analysis courses have been unfamiliar withinvariance arguments, although they usually meet them in subsequentcourses. For this reason I have long felt that an introduction to invarianttests in a multivariate text would certainly not be out of place

viii Prejulte 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 differential 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 estimation 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

PrefaceixChapter 7 is where this book departs most significantly from others onmultivariate statistical theory. Here the groundwork is laid for studying thenoncentral distribution theory needed in subsequent chapters, where theemphasis is on testing problems in standard multivariate procedures.Zonalpolynomials 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, oflikelihood ratio statistics used for testing standard hypotheses about covari-ance matrices and mean vectors. An attempt is also made here to explainwhat happens if these tests are used and the underlying distribution isnon-normal.Chapter 9deals with theprocedureknown as principal compo-nents, where much attention is focused on the Jatent roots of the samplecovariance matrix.Asymptotic distributions of these roots are obtained andare used in various inference problems. Chapter l0 studies the multivariategeneral linear model and the distribution of latent roots and functions ofthem used for testing the general linear hypothesis. An introduction todiscriminant analysis is also included here, although the coverage is ratherbrief.Finally,Chapter Il deals with the problem of testing independencebetween a number of sets of variables and also with canonical correlationanalysis.The choice of the material covered is, of course, extremely subjective andlimited 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 thetopics included form the core of a reasonable course in classicalmultivariate analysis. Areas which are not covered in the text include factoranalysis, multiple time series, multidimensional scaling, clustering, anddiscrete multivariate analysis. These topics have grown so large that thereare now separate books devoted to each. The coverage of classification anddiscriminant analysis also is not very extensive, and no mention is made ofBayesian approaches; these topics have been treated in depth by Andersonand by Kshirsagar, Multivariate Analysis, and Srivastava and Khatri, AnIntroduction to Multivariate Statistics, and a person using the current workas a textmay wishto supplement it with material fromthesereferences.Thisbookhasbeen plannedasa text for atwo-semester courseinmultivariate statistical analysis. By an appropriate choice of topics it canalso be used in a one-semester course.One possibility is to cover Chapters l,2,3,5,and possibly 6, and those sections of Chapters 8,9, 10 and 11 whichdo not involve noncentral distributions and consequently do not utilize thetheory developed in Chapter 7. The book is designed so that for the mostpart these sections can be easily identified and omitted if desired. Exercisesare provided at the end of each chapter. Many of these deal with points

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 introduced, 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 covariance 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 components, 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 mentioned 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

Prefacewhich are alluded to in the text but left unproved. A few words are also inorder concerning the Bibliography.I have not felt it necessary to cite thesource of every result included here. Many of the original results due to suchpeopleas Wilks,Hotelling,Fisher,Bartlett,Wishart,andRoyhavebecomeso well known that they are now regarded as part of the folklore ofmultivariate analysis. T. W. Anderson's book provides an extensive bib-liography of work prior to 1958, and my references to early work are in-discriminate at best. I have tried to be much more careful concerningreferences to the more recent work presented in this book,particularly inthe area of distribution theory. No doubt some references have been missed,but I hope that the number of these is small. Problems which have beentaken from the literature are for the most part not referenced unless theproblem is especially complex or the reference itself develops interestingextensions and applications that theproblem does not cover.This book owes much to many people. My teachers, A.T. James andA.G.Constantine,have had a distinctiveinfluence on me and their ideasare in evidence throughout, and especially in Chapters 2, 3, 7, 8, 9, 10, andIl.I am indebted to them both.Many colleagues and students havereadcriticized, and corrected various versions of the manuscript. J. A. Hartiganread the first four chapters,and Paul Sampson used parts of the first ninechapters for a course at the University of Chicago; I am grateful to both fortheir extensivecomments,corrections,and suggestions.Numerous othershave also helped to weed out errors and have influenced the final version;especially deserving of thanks are D. Bancroft, W. J. Glynn, J. Kim, M.Kramer, R. Kuick, D. Marker, and J. Wagner. It goes without saying thatthe responsibility for all remaining errors is mine alone. I would greatlyappreciate being informed ahout any that are found, large and small.A number of peopletackled the unenviabletask of typingvarious partsand revisionsof themanuscript.For theirexcellentwork and theirpatiencewith my handwriting I would like to thank Carol Hotton, Terri LomaxHunter, Kelly Kane, and Deborah Swartz.ROBBJ.MUIRHEADAnnArbor,MichigunFebruary1982

x Prefure which are alluded to in the text but left unproved. A few words are also in order concerning the Bibliography. I have not felt it necessary to cite the source of every result included here. Many of the original results due to such people as Wilks, Hotelling, Fisher, Bartlett, Wishart, and Roy have become so well known that they are now regarded as part of the folklore of multivariate analysis. T. W. Anderson’s book provides an extensive bibliography of work prior to 1958, and my references to early work are indiscriminate at best, I have tried to be much more careful concerning references to the more recent work presented in this book, particularly in the area of distribution theory. No doubt some references have been missed, but I hope that the number of these is small. Problems which have been taken from the literature are for the most part not referenced unless the problem is especially complex or the reference itself develops interesting extensions and applications that the problem does not cover. This book owes much to many people. My teachers, A. T. James and A. G. Constantine, have had a distinctive influence on me and their ideas are in evidence throughout, and especially in Chapters 2, 3, 7, 8, 9, 10, and 11. I am indebted to them both. Many colleagues and students have read, criticized, and corrected various versions of the manuscript. J. A. Hartigan read the first four chapters, and Paul Sampson used parts of the first nine chapters for a course at the University of Chicago; I am grateful to both for their extensive comments, corrections, and suggestions. Numerous others have also helped to weed out errors and have influenced the final version; especially deserving of thanks are D. Bancroft, W. J. Glynn, J. Kim, M. Kramer, R. Kuick, D. Marker, and J. Wagner. It goes without saying that the responsibility for all remaining errors is mine alone. I would greatly appreciate being informed about any that are found, large and small. A number of people tackled the unenviable task of typing various parts and revisions of the manuscript. For their excellent work and their patience with my handwriting 1 would like to thank Carol Hotton, Terri Lomax Hunter, Kelly Kane, and Deborah Swartz. RODD J. MUIRMEAD Ann Arbor, Mirhrgun February I982