An Introduction to MultivariateStatistical Analysis (3ed)

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An Introduction to MultivariateStatistical Analysis (3ed)
Contents Presa the Third Editi Preface to the Second Editi p1 E NrS I 1 Introduction 1.1.Mu nalysis, 1 1.2. The multivariate Normal Distribution, 3 The Multivariate Normal Distr 2. 1. Introduction 2, 2. notic te distributi 2. 3. The Multivariate Normal Distribution, 13 2.4. The Distribution of Linear Combinations of Normally Distributed variates: Independ ginal distrib .5. Conditional Distributions and Multiple Correl 2.6. The Characteristic Function: Moments, 41 2.7. Elliptically Contoured Distributions, 47 Problems, 56 3 Estimation of the Mean Vector and the Covariance matrix 66 3.1. Introduction 3.2. The Maximum Likelihood estimators u[ the Mcan Vector 6.4. Classitication into One of Two Known Multivariate normal and the Covariance matrix. 67 Populations, 215 3. The Distribution of the sample mean vec 6, 5. Classification into o wo Multivariate normal ing the mcan when th ariance matrix ls ted. 219 KI 6.6. Probabilities of misclassification 227 3.4. Theoretical Properties of Estimators of the Mean vector、83 6.7. Classification into One of Several Populations, 233 3.5. Improved Estimation of the Mcan, 91 6.8. Classification into One of Several Multivariate normal 3.6. Elliptically c 101 6.9. An Example of Classifi to one of several Problems. 108 Multivariate Normal populations, 240 610. Classificati 4 The Distributions and Uses of Sample Correlation Coefficients 115 Populations with Unequal Covariance Matrices, 242 Problems 248 4. 1. Introduction. 115 4. 2. Currelation Coefficient of a Bivariate Sample, 116 43. Partial Correlation Cocfficients: Conditional 7 The Distribution of the Sample CovariE nce Matrix and the Distributions. 130 Sample generalized variance 4. 4. The Multiple Corrclation Coefficient, 144 4.5. Elliptically Contoured Distributions, 158 7.1. Introductio Problems. 16.3 7. 2. The wishart distribution 252 7.3. Some Properties of the wishart Distribution, 258 5 The generalized T Statistic 170 7.4. Cochran's Theorem. 262 7.5. The generalized v: e.264 5.1. Introduction, 170 7.6. Distribution of the Set of Correlation Coefficients When 5.2. Derivation of the Generalized T2 Statistic and Its the Population Covariance Matrix Is Diagonal, 270 Distribution 171 7. 7. The Inverted wishart Distribution and Bayes Estimation of .3. Uses of the T--Statistic, I the co 72 5. 4. The Distribution of T<under Alternative Hypotheses 7.8. Improved Estimation of the Covariance Matrix, 276 The Power Function, 18 7.g. Elliptically Contoured Distributions, 282 5.5. The Two-Sample Problem with Unequal Covariance Problems, 285 Matrices. 187 6. Some Opiimal properties of the T-Test, 190 8 Testing the General Linear Hypothesis; Multivariate analysis 5.7. Elliptically Contoured Distributions, 199 of variance Problems. 201 8.1. Introduction. 29 1 6 Classification of Observations 207 8.2. Estimators of Parameters in Multivariate Linear Regi'es 292 6. 1. The Problem of Classification, 207 8.3. Likelihood Ratio Criteria for Testing Linear Hypotheses 6. 2. Standards of Good Classification 208 about ro Procedures (br Cliissificalito into Onc of 'T'wo Populations 8. 4. The Distribution of the Likelihood Ratio Criterion when Probability distributions, 211 the Hypothesis Is CONTENTS CONTENTS asymptotic Expansion of the distrib 10.5. Asymptotic Expansions of the Distributions of the lihood ratio Criterion, 316 Criteria, 424 S.6. Other Criteria for Testing the Linear Hypothesis, 326 10.6. The Case of Two Populations, 427 8.7. Tests of Hypotheses about matrices of regres 10.7. Testing the Hypothesis That a Covariance M Coefficients and Confidence Regions, 337 Is Proportional to a Given Matrrix; The Sphericity S.S. Testing Equality of Means of Several Normal Distributions Test. 431 with common covariance matrix. 342 10.8. Testing the Hypothesis That a Covariance Matrix Is 8.9. Multivariate Analysis of variance, 346 Equal to a Given Matrix, 438 8. 10. Some Optimal properties of Tests, 353 10.9. Testing the Hypothesis That a mean Vector and a 8. 11. Elliptically Contoured Distributions, 370 Covariance Matrix Are Equal to a Given Vector ane Problems. 374 10.10. Admissibility of Tests, 446 9 Testing Independence of Sets of variates 10.11. Elliptically Contoured Distributions, 449 Problems. 454 9. 1. lntroduction 381 9. 2. The Likelihood Ratio Critcrion for Testing independence 11 Principal Components 459 93, The distribution of the likelhood ratio criterion when 11.1. Introduction, 459 the Null hypothesis Is True, 386 11.2. Definition of Principal Components in the 9. 4. An ASy mptotic Expansion of the Distribution of the Population, 460 Likelihood ratio criterion 390 11.3. Maximum Likelihood Estimators of the Principal 9. 5. Other Criteria, 39 Components and Their variances, 467 9,6. Step-Down Procedures, 393 11. 4. Computation of the Maximum Likelihood Estimates of 97. An Example, 396 ponents, 469 9. 8. The Case of Two Sets of Variates, 397 11.5. An Example, 471 9. 9. Admissibility of the likelihood Ratio Test, 401 11.6. Statistical Inference. 473 9. Independence of Sets, 40)2 10. Monotonicity of Power Funct 11.7. Testing Hypotheses about the Characteristic Roots of a Covariance matrix. 478 9.11. Elliptically Contoured Distributions, 40 11.8. Elliptically Contoured Distributions, 482 Problems. 40 Problems, 483 10 Testing Hypotheses of Equality of Covariance Matrices and 12 Canonical Correlations and Canonical variables Equality of Mean Vectors and Covariance Matrices 411 12.1. Introduction. 487 10.1. Introduction. 411 12,2. Canonical Correlations and variates in the 10. 2. Criteria for Testing Equality of Several Covariance Population, 488 Matrices. 412 12.3. Estimation of Canonical Correlations and Variates, 498 10.3. Criteria for Testing That Several Normal Distributions 12. 4. Statistical Inference, 503 Are identical、415 12.5.AnE 10.4. Distributions of the Criteria, 417 12.6. Linearly related Expected values, 508 CONTENTS CONTENTS 12.7. Reduced Rank Regression, 514 A3. Partitioned Vectors and Matrices. 635 12.8. Simultaneous Equations Models, 515 A4. Some Miscellaneous Results 639 Problems. 526 A,5. Gram-Schmidt Orthogonalization and the solntion of Linear Equations, 647 13 The Distributions of Characteristic Roots and Vectors 528 Appendix B Tables 651 13.1. Introduction. 528 13. 2. The case of Two wishart Matrices. 529 B.1.wilks'Likelihood Criterion: Factors C(p, m, M)to 13.3. The Case of One Nonsingular Wishart Matrix, 538 Adjust to Xp, m, where M=n-p+ 1, 651 134.Ca al Correlations, 543 B 2. Tables of Significance Points for the Lawley-Hotelling 13.5. Asymptotic Distributions in the Case of One Wishart TIace Test, 657 B.3. Tables of Significance Points for the 13.6. Asymptotic Distributions in the Case of Two Wishart Bartlett-Nanda Pillai Trace Test. 673 Matrices. 549 B 4. Tables of significance Points for the roy maximum Root 13. 7. Asymptotic Distribution in a Regression Model, 555 13.8. Elliptically Contoured Distributions, 563 B 5. Significance Points for the Modified Likelihood ratio Problems. 567 Test of Equality of Covariance Matrices Based on Equal Sample Sizes, 681 ctor Analysi 569 B.6. Correction Factors for Significance Points for the Sphericity Test, 683 14.1. Introduction. 569 B 7. Significance Points for the Modified Likelihood Ratio 14.2. The Model, 570 rest习=习n685 14.3. Maximum Likelihood Estimators for Random Orthogonal Factors, 576 References 687 14. 4. Estimation for Fixed Factors. 586 Inde 713 14.5. Factor Interpretation and Transformation, 587 14. 6. Esti for Identification by Specified Ze 14,7. Estimation of Factor S Proble 15 Patterns of Dependence; Graphical Models 595 15.1. Introduction,. 59 15.3. Directed graphs, 604 5.4. Chain Graphs 610 15.5. Statistical Infe 613 Appendix A Matrix Theory A 1. Definition of a matrix and n matrices, 624 A. 2. Characteristic Roots and Vectors, 631 Preface to the Third edition For some forty years the first and second editions of this book have been used by students to acquire a basic knowledge of the theory and methods of multivariate statistical analysis. The book has also served a wider com f statisticians in furthering their understanding and proficiency in this field Since the second edition was published, multivariate analysis has be developed and extended in many directions. Rather than attempting to cover, thie enlarged I have elected to elucidate several aspects that are particularly interesting and useful for methodology and comprehen SIon. Earlier editions included some methods that could be carried out on an dding machine! In the twenty-first century, however, computational tech- niques have become so highly developed and improvements come so rapidly that it is impossible to include all of the relevant methods in a volume on the general mathematical theory. Some aspects of statistics exploit computational powe ogles, The definition of multivariate statistics implies the treatment of variables that are interrelated. Several chapters are devoted to measures of correlation and tests of independence. A new chapter, "Patterns of Dependence; Graph ical Models " has been added. A So-called graphical model is a set of vertices gei suggesting dependences between variables. The algebra of such graphs is an outgrowth and development of path analysis and the study of causal chains a graph may represent a sequence in time or logic and may suggest causation of one set of variables by another set. Another new topic systematically presented in the third edition is that of elliptically The multivariate which is characterized by the mean vector and covariance matrix, has a limitation that the fourth-order moments of the variables are determined by the first- and second-order moments. The class of elliptically contoured PREFACE TO THE THIRD EDITTON distribution relaxes this restriction. A density in this class has contours of equal density which are ellipsoids as does a normal density but the set of fourth-orde This topic is expounded by the addition of sections to appropriate chapters Reduced rank regression 13 provides a method of reducing the number of regression coefficients to be estimated in the regression of one set of variables to another. This approach includes the limited-information maximum-likelihood estimator of an equation in a simul- Preface to the Second edition taneous equations model The preparation of the third edition has been benefited by advice and comments of readers of the first and second editions as well as by reviewers of the current revision. In addition to readers of the earlier editions listed in those prefaces I want to thank Michael Perlman and Kathy Richards for their assistance in getting this manuscript ready Twenty-six years have passed since the first edition of this book was pub lished. During that time great advances have been made in multivariate T.W.. ANDERSON statistical analysis--particularly in the areas treated in that volume. This new edition purports to bring the original edition up to date by substanti revision, rewriting, and additions. The basic approach has been maintained Feby ly rigorous development of statistical methods for observations consisting of several measurements or characteristics of each subject and a study of their properties, The general outline of topics has been The method of maximum likelihood has been augmented by other consid erations. In point estimation of the mean vector and covariance matrix ves to the maximum likelihood estimators that are better with Bay lihood ratio tests have been pplemented by other invariant procedures, New results on distributions and asymptotic distributions are given; some significant points are tabulated Properties of these procedures, such as power functions, admissibility, uni asedness, and monotonicity of power functions, are studied, Simultaneous confidence intervals for means and covariances are developed. a chapter on places the chapter sketching miscell p linear functional relationships, are introduced. Additional problems present rther res It is impossible to cover all relevant material in this book; what seems most important has been included. For a comprehensive listing of papers til 1966 and books until 1970 the reader is referred to A Bibliography of Multivariate Statistical Analysis by A Das gupt: 1972) Further references can be found in Multivariate Analysis: A Selected and X1 XAU PREFACE TO THE SECOND EDITION Abstracted Bibliography, 1957-1972 by Subrahmaniam and Subrahmaniam (1973) I am in debt to many students, colleagues, and friends for their suggestions and assistance; they include Yasuo Amemiya, James Berger, Byoung-Seon uIr Somesh Das Gupta, Kai-1 Gene Golub, Aaron Han, Takeshi Hayakawa, Jogi Henna, Huang Hsu, Fred Huffer, Mituaki Huzii, Jack Kiefer, Mark Knowles, Sue Leurgans, Alex McMillan, Masashi No, Ingram Olkin, Kartik PateL, Michael Periman, Allen Preface to the first Edition Sampson, Ashis Sen Gupta, Andrew Siegel, Charles Stein, Patrick Strou Akimichi Takemura, Joe Verducci, Marlos Viana, and Y. Yajima. I was helped in preparing the manuscript by Dorothy Anderson, Alice lundin ny Schwartz, and Pat Struse. Special thanks go to Johanne Thiffault and George P. H. Styan for their precise attention. Support was contributed by the army Research Office, the National Science Foundation, the Office of This book has been designed primarily as a text for a two-semester course in Naval Research, and IBM Systems Research Institut multivariate statistics. It is hoped that the book will also serve as an Seven tables of significance points are given in Appendix b to facilitate introduction to many topics in this area to statisticians who are not students carrying out test procedures. Tables 1, 5, and 7 are Tables 47, 50, and 53 and will be used as a reference by other statisticians. respectively, of Biometrika Tables for Statisticians, Vol. 2, by E. S.Pearson For several years the book in the form of dittoed notes has been used in a and H.O. Hartley; permission of the Biometrika Trustees is hereby acknow- two-semester sequence of graduate courses at Columbia University, the first edged. Table 2 is made up from three tables prepared by A. W. Davis and ix chapters constituted the text for the first semester, emphasizing correla published in Biometrika( 1970a), Annals of the Institute of Statistica! Mathe- tion theory. It is assumed that the reader is familiar with the usual theory of Platics(1970b)and Communications in Statistics, B Simulation and computa univariate statistics, particularly methods based on the univariate normal tion( 1980). Tables 3 and 4 are Tables 6. 3 and 6.4, respectively, of Concise distribution. A knowledge of matrix algebra is also a prerequisite; however, Statistical Tables, edited by Ziro Yamauti(1977)and published by the an appendix on this topic has been included Japanese Standards Association; this book is a concise version of Statisticai It is hoped that the more basic and important topics are treated here, Tables and Formulas with Computer ApplicationS, JSA-1972. Table 6 is Table 3 though to some extent the coverage is a matter of taste. Some of the more of The Distribution of the Sphericity Tesf Criterion, ARL 72-0154, by B. N recent and advanced developments are only briefly touched on in the late Nagarsenker and K. C. S. Pillai, Acrospace Rcscarch Laboratorics(1972) chapter The author is indebted to the authors and publishers listed above for The method of maximum likelihood is used to a large extent. This leads to permission to reproduce these tables reasonable procedures; in some cases it can be proved that they are optimal. In many situations, however, the theory of desirable or optimum procedures T W ANDERSON IS lacking Over the years this manuscript has been developed, a number of students Stanford. Califomia and colleagues have been of considerable assistance. Allan Birnbaum, Harold June /984 Hotelling, Jacob Horowitz, Howard Levene, Ingram OIkin, Gobind Seth, Charles Stein, and Henry Teicher are to be mentioned particularly, Acknowl dgements are also due to other members of the Graduate mathematical PREFACE TO THE FIRST EDITION Statistics Society at Columbia University for aid in the preparation of the The prep2 this ipt was sup ported in part by the Office of Naval Research CHAPTER 1 ①.W. ANDERSON ter for Adoanced study Introduction stanford, california December 957 1.1. MULTIVARIATE STATISTICAL ANALYSIS Multivariate statistical analysis is concerned with data that consist of sets of measurements on a number of individuals or objects. The sample data may be heights and weights of some individuals drawn randomly from a popula- tion of school children in a given city, of the statistical treatment may be de on a colle h as lengths and widths of petals and lengths and widths of sepals of iris plants taken from two species, or on may study the scores on batteries of mental tests administered to a number of The measurements made on a single individual can be assembled into a column vector. We think of the entire vector as an observation from a multivariate population or distribution. When the individual is drawn ran lomly, we consider the vector as a random vector with a distribution or probability law describing that population. The set of observations on all individuals in a sample constitutes a sample of vectors, and the vectors set side by side make up the matrix of observations. The data to be analyzee then are thought of as displayed in a matrix or in Several matrices. We shall see that it is helpful in visualizing the data and understanding the methods to think of each observation vector as constituting a point in a Euclidean sp dinate corresponding to a mei able. Indeed an early step in the statist lysis is plotting the data; sir When data are listed on paper by individual, it is natural to print the of the table; then one individual corre to operate algebraically with column vectors, we have chosen to treat observations in terms of column vectors. (In practice, the basic data set may well be on cards, tapes, or disks) An Introduction to Multivariate Statistical Analysis, Third Edition. By T.w.Anderson ISBN 0-471-36091-0 Copyright ht 2003 John Wiley Sons, in 【实例截图】
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