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An introduction to multivariate statistical analysis /
T.W. Anderson.
2nd ed. --
New York : Wiley, 1984.
xvii, 675 p. : ill.
More Details
New York : Wiley, 1984.
general note
Includes bibliographical references and index.
catalogue key
A Look Inside
About the Author
Author Affiliation
T.W. Anderson Professor Emeritus of Statistics and Economics at Stanford University, earned his PhD in mathematics at Princeton University. Anderson is a member of the National Academy of Sciences and a Fellow of the Institute of Mathematical Statistics, the American Statistical Association, the Econometric Society, and the American Academy of Arts and Sciences
Flap Copy
An Introduction to Multivariate Statistical Analysis, 2nd Edition is a major updating of a work widely regarded as the standard, authoritative text in the field. It provides students and practicing statisticians with the latest theory and methods, plus the most important developments that have occurred over the past 25 years. As in the first edition, the text provides a mathematically rigorous development of the statistical methods used to analyze multivariate data. While maximum likelihood estimators have been principal tools of multivariate statistical analysis, this book introduces alternatives that are better suited for certain loss functions, such as Stein and Bayes estimators. Likelihood ratio tests have been supplemented by other invariant procedures. New results on distributions are given and some significance points are tabulated. Properties of these procedures such as power functions, admissibility, unbiasedness, and monotonicity of power functions are covered, and simultaneous confidence intervals for means and covariances are studied. Other new topics introduced in this edition include simultaneous equations models and linear functional relationships, with 50% more problems than in the previous edition.
Long Description
Multivariate Statistical Simulation Mark E. Johnson For the researcher in statistics, probability, and operations research involved in the design and execution of a computer-aided simulation study utilizing continuous multivariate distributions, this book considers the properties of such distributions from a unique perspective. With enhancing graphics (three-dimensional and contour plots), it presents generation algorithms revealing features of the distribution undisclosed in preliminary algebraic manipulations. Well-known multivariate distributions covered include normal mixtures, elliptically assymmetric, Johnson translation, Khintine, and Burr-Pareto-logistic. 1987 (0 471-82290-6) 230 pp. Aspects of Multivariate Statistical Theory Robb J. Muirhead A classical mathematical treatment of the techniques, distributions, and inferences based on the multivariate normal distributions. The main focus is on distribution theory-both exact and asymptotic. Introduces three main areas of current activity overlooked or inadequately covered in existing texts: noncentral distribution theory, decision theoretic estimation of the parameters of a multivariate normal distribution, and the uses of spherical and elliptical distributions in multivariate analysis. 1982 (0 471-09442-0) 673 pp. Multivariate Observations G. A. F. Seber This up-to-date, comprehensive sourcebook treats data-oriented techniques and classical methods. It concerns the external analysis of differences among objects, and the internal analysis of how the variables measured relate to one another within objects. The scope ranges from the practical problems of graphically representing high dimensional data to the theoretical problems relating to matrices of random variables. 1984 (0 471-88104-X) 686 pp.
Main Description
Perfected over three editions and more than forty years, this field- and classroom-tested reference: * Uses the method of maximum likelihood to a large extent to ensure reasonable, and in some cases optimal procedures. * Treats all the basic and important topics in multivariate statistics. * Adds two new chapters, along with a number of new sections. * Provides the most methodical, up-to-date information on MV statistics available.
Table of Contents
Preface to the Third Editionp. xv
Preface to the Second Editionp. xvii
Preface to the First Editionp. xix
Introductionp. 1
Multivariate Statistical Analysisp. 1
The Multivariate Normal Distributionp. 3
The Multivariate Normal Distributionp. 6
Introductionp. 6
Notions of Multivariate Distributionsp. 7
The Multivariate Normal Distributionp. 13
The Distribution of Linear Combinations of Normally Distributed Variates; Independence of Variates; Marginal Distributionsp. 23
Conditional Distributions and Multiple Correlation Coefficientp. 33
The Characteristic Function; Momentsp. 41
Elliptically Contoured Distributionsp. 47
Problemsp. 56
Estimation of the Mean Vector and the Covariance Matrixp. 66
Introductionp. 66
The Maximum Likelihood Estimators of the Mean Vector and the Covariance Matrixp. 67
The Distribution of the Sample Mean Vector; Inference Concerning the Mean When the Covariance Matrix Is Knownp. 74
Theoretical Properties of Estimators of the Mean Vectorp. 83
Improved Estimation of the Meanp. 91
Elliptically Contoured Distributionsp. 101
Problemsp. 108
The Distributions and Uses of Sample Correlation Coefficientsp. 115
Introductionp. 115
Correlation Coefficient of a Bivariate Samplep. 116
Partial Correlation Coefficients; Conditional Distributionsp. 136
The Multiple Correlation Coefficientp. 144
Elliptically Contoured Distributionsp. 158
Problemsp. 163
The Generalized T[superscript 2]-Statisticp. 170
Introductionp. 170
Derivation of the Generalized T[superscript 2]-Statistic and Its Distributionp. 171
Uses of the T[superscript 2]-Statisticp. 177
The Distribution of T[superscript 2] under Alternative Hypotheses; The Power Functionp. 185
The Two-Sample Problem with Unequal Covariance Matricesp. 187
Some Optimal Properties of the T[superscript 2]-Testp. 190
Elliptically Contoured Distributionsp. 199
Problemsp. 201
Classification of Observationsp. 207
The Problem of Classificationp. 207
Standards of Good Classificationp. 208
Procedures of Classification into One of Two Populations with Known Probability Distributionsp. 211
Classification into One of Two Known Multivariate Normal Populationsp. 215
Classification into One of Two Multivariate Normal Populations When the Parameters Are Estimatedp. 219
Probabilities of Misclassificationp. 227
Classification into One of Several Populationsp. 233
Classification into One of Several Multivariate Normal Populationsp. 237
An Example of Classification into One of Several Multivariate Normal Populationsp. 240
Classification into One of Two Known Multivariate Normal Populations with Unequal Covariance Matricesp. 242
Problemsp. 248
The Distribution of the Sample Covariance Matrix and the Sample Generalized Variancep. 251
Introductionp. 251
The Wishart Distributionp. 252
Some Properties of the Wishart Distributionp. 258
Cochran's Theoremp. 262
The Generalized Variancep. 264
Distribution of the Set of Correlation Coefficients When the Population Covariance Matrix Is Diagonalp. 270
The Inverted Wishart Distribution and Bayes Estimation of the Covariance Matrixp. 272
Improved Estimation of the Covariance Matrixp. 276
Elliptically Contoured Distributionsp. 282
Problemsp. 285
Testing the General Linear Hypothesis; Multivariate Analysis of Variancep. 291
Introductionp. 291
Estimators of Parameters in Multivariate Linear Regressionp. 292
Likelihood Ratio Criteria for Testing Linear Hypotheses about Regression Coefficientsp. 298
The Distribution of the Likelihood Ratio Criterion When the Hypothesis Is Truep. 304
An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterionp. 316
Other Criteria for Testing the Linear Hypothesisp. 326
Tests of Hypotheses about Matrices of Regression Coefficients and Confidence Regionsp. 337
Testing Equality of Means of Several Normal Distributions with Common Covariance Matrixp. 342
Multivariate Analysis of Variancep. 346
Some Optimal Properties of Testsp. 353
Elliptically Contoured Distributionsp. 370
Problemsp. 374
Testing Independence of Sets of Variatesp. 381
Introductionp. 381
The Likelihood Ratio Criterion for Testing Independence of Sets of Variatesp. 381
The Distribution of the Likelihood Ratio Criterion When the Null Hypothesis Is Truep. 386
An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterionp. 390
Other Criteriap. 391
Step-Down Proceduresp. 393
An Examplep. 396
The Case of Two Sets of Variatesp. 397
Admissibility of the Likelihood Ratio Testp. 401
Monotonicity of Power Functions of Tests of Independence of Setsp. 402
Elliptically Contoured Distributionsp. 404
Problemsp. 408
Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matricesp. 411
Introductionp. 411
Criteria for Testing Equality of Several Covariance Matricesp. 412
Criteria for Testing That Several Normal Distributions Are Identicalp. 415
Distributions of the Criteriap. 417
Asymptotic Expansions of the Distributions of the Criteriap. 424
The Case of Two Populationsp. 427
Testing the Hypothesis That a Covariance Matrix Is Proportional to a Given Matrrix; The Sphericity Testp. 431
Testing the Hypothesis That a Covariance Matrix Is Equal to a Given Matrixp. 438
Testing the Hypothesis That a Mean Vector and a Covariance Matrix Are Equal to a Given Vector and Matrixp. 444
Admissibility of Testsp. 446
Elliptically Contoured Distributionsp. 449
Problemsp. 454
Principal Componentsp. 459
Introductionp. 459
Definition of Principal Components in the Populationp. 460
Maximum Likelihood Estimators of the Principal Components and Their Variancesp. 467
Computation of the Maximum Likelihood Estimates of the Principal Componentsp. 469
An Examplep. 471
Statistical Inferencep. 473
Testing Hypotheses about the Characteristic Roots of a Covariance Matrixp. 478
Elliptically Contoured Distributionsp. 482
Problemsp. 483
Canonical Correlations and Canonical Variablesp. 487
Introductionp. 487
Canonical Correlations and Variates in the Populationp. 488
Estimation of Canonical Correlations and Variatesp. 498
Statistical Inferencep. 503
An Examplep. 505
Linearly Related Expected Valuesp. 508
Reduced Rank Regressionp. 514
Simultaneous Equations Modelsp. 515
Problemsp. 526
The Distributions of Characteristic Roots and Vectorsp. 528
Introductionp. 528
The Case of Two Wishart Matricesp. 529
The Case of One Nonsingular Wishart Matrixp. 538
Canonical Correlationsp. 543
Asymptotic Distributions in the Case of One Wishart Matrixp. 545
Asymptotic Distributions in the Case of Two Wishart Matricesp. 549
Asymptotic Distribution in a Regression Modelp. 555
Elliptically Contoured Distributionsp. 563
Problemsp. 567
Factor Analysisp. 569
Introductionp. 569
The Modelp. 570
Maximum Likelihood Estimators for Random Orthogonal Factorsp. 576
Estimation for Fixed Factorsp. 586
Factor Interpretation and Transformationp. 587
Estimation for Identification by Specified Zerosp. 590
Estimation of Factor Scoresp. 591
Problemsp. 593
Patterns of Dependence; Graphical Modelsp. 595
Introductionp. 595
Undirected Graphsp. 596
Directed Graphsp. 604
Chain Graphsp. 610
Statistical Inferencep. 613
Matrix Theoryp. 624
Definition of a Matrix and Operations on Matricesp. 624
Characteristic Roots and Vectorsp. 631
Partitioned Vectors and Matricesp. 635
Some Miscellaneous Resultsp. 639
Gram-Schmidt Orthogonalization and the Solution of Linear Equationsp. 647
Tablesp. 651
Wilks' Likelihood Criterion: Factors C(p, m, M) to Adjust to x[superscript 2 subscript p, m], where M = n - p + 1p. 651
Tables of Significance Points for the Lawley-Hotelling Trace Testp. 657
Tables of Significance Points for the Bartlett-Nanda-Pillai Trace Testp. 673
Tables of Significance Points for the Roy Maximum Root Testp. 677
Significance Points for the Modified Likelihood Ratio Test of Equality of Covariance Matrices Based on Equal Sample Sizesp. 681
Correction Factors for Significance Points for the Sphericity Testp. 683
Significance Points for the Modified Likelihood Ratio Test [Sigma] = [Sigma subscript 0]p. 685
Referencesp. 687
Indexp. 713
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