An introduction to multivariate statistical analysis [3 ed.] 0471360910, 9661997470, 9073115027, 2236956576, 2591811717, 4810751104, 1110538103

Table of contents :
Contents......Page 5
Preface to the Third Edition......Page 13
Preface to the Second Edition......Page 15
Preface to the First Edition......Page 17
1 Introduction......Page 19
1.1. Multivariate Statistical Analysis......Page 19
1.2. The Multivariate Normal Distribution......Page 21
2 The Multivariate Normal Distribution......Page 24
2.1. Introduction......Page 24
2.2. Notions of Multivariate Distributions......Page 25
2.3. The Multivariate Normal Distribution......Page 31
2.4. The Distribution of Linear Combinations of Normally Distributed Variates; Independence of Variates; Marginal Distributions......Page 41
2.5. Conditional Distributions and Multiple Correlation Coefficient......Page 51
2.6. The Characteristic Function; Moments......Page 59
2.7. Elliptically Contoured Distributions......Page 65
Problems......Page 74
3 Estimation of the Mean Vector and the Covariance Matrix......Page 84
3.1. Introduction......Page 84
3.2. The Maximum Likelihood Estimators of the Mean Vector and the Covariance Matrix......Page 85
3.3. The Distribution of the Sample Mean Vector; Inference Concerning the Mean When the Covariance Matrix Is Known......Page 92
3.4. Theoretical Properties of Estimators of the Mean Vector......Page 101
3.5. Improved Estimation of the Mean......Page 109
3.6. Elliptically Contoured Distributions......Page 119
Problems......Page 126
4 The Distributions and Uses of Sample Correlation Coefficients......Page 133
4.1. Introduction......Page 133
4.2. Correlation Coefficient of a Bivariate Sample......Page 134
4.3. Partial Correlation Coefficients; Conditional Distributions......Page 154
4.4. The Multiple Correlation Coefficient......Page 162
4.5. Elliptically Contoured Distributions......Page 176
Problems......Page 181
5 The Generalized T^2-Statistic......Page 188
5.1. Introduction......Page 188
5.2. Derivation of the Generalized T^2-Statistic and Its Distribution......Page 189
5.3. Uses of the T^2-Statistic......Page 195
5.4. The Distribution of T^2 under Alternative Hypotheses; The Power Function......Page 203
5.5. The Two-Sample Problem with Unequal Covariance Matrices......Page 205
5.6. Some Optimal Properties of the T^2-Test......Page 208
5.7. Elliptically Contoured Distributions......Page 217
Problems......Page 219
6 Classification of Observations......Page 225
6.1. The Problem of Classification......Page 225
6.2. Standards of Good Classification......Page 226
6.3. Procedures of Classification into One of Two Populations with Known Probability Distributions......Page 229
6.4. Classification into One of Two Known Multivariate Normal Populations......Page 233
6.5. Classification into One of Two Multivariate Normal Populations When the Parameters Are Estimated......Page 237
6.6. Probabilities of Misclassification......Page 245
6.7. Classification into One of Several Populations......Page 251
6.8. Classification into One of Several Multivariate Normal Populations......Page 255
6.9. An Example of Classification into One of Several Multivariate Normal Populations......Page 258
6.10. Classification into One of Two Known Multivariate Normal Populations with Unequal Covariance Matrices......Page 260
Problems......Page 266
7 The Distribution of the Sample Covariance Matrix and the Sample Generalized Variance......Page 269
7.1. Introduction......Page 269
7.2. The Wishart Distribution......Page 270
7.3. Some Properties of the Wishart Distribution......Page 276
7.4. Cochran's Theorem......Page 280
7.5. The Generalized Variance......Page 282
7.6. Distribution of the Set of Correlation Coefficients When the Population Covariance Matrix Is Diagonal......Page 288
7.7. The Inverted Wishart Distribution and Bayes Estimation of the Covariance Matrix......Page 290
7.8. Improved Estimation of the Covariance Matrix......Page 294
7.9. Elliptically Contoured Distributions......Page 300
Problems......Page 303
8 Testing the General Linear Hypothesis; Multivariate Analysis of Variance......Page 309
8.1. Introduction......Page 309
8.2. Estimators of Parameters in Multivariate Linear Regression......Page 310
8.3. Likelihood Ratio Criteria for Testing Linear Hypotheses about Regression Coefficients......Page 316
8.4. The Distribution of the Likelihood Ratio Criterion When the Hypothesis Is True......Page 322
8.5. An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterion......Page 334
8.6. Other Criteria for Testing the Linear Hypothesis......Page 344
8.7. Tests of Hypotheses about Matrices of Regression Coefficients and Confidence Regions......Page 355
8.8. Testing Equality of Means of Several Normal Distributions with Common Covariance Matrix......Page 360
8.9. Multivariate Analysis of Variance......Page 364
8.10. Some Optimal Properties of Tests......Page 371
8.11. Elliptically Contoured Distributions......Page 388
Problems......Page 392
9 Testing Independence of Sets of Variates......Page 399
9.1. Introduction......Page 399
9.2. The Likelihood Ratio Criterion for Testing Independence of Sets of Variates......Page 399
9.3. The Distribution of the Likelihood Ratio Criterion When the Null Hypothesis Is True......Page 404
9.4. An Asymptotic Expansion of the Distribution of the Likelihood Ratio Criterion......Page 408
9.5. Other Criteria......Page 409
9.6. Step-Down Procedures......Page 411
9.7. An Example......Page 414
9.8. The Case of Two Sets of Variates......Page 415
9.9. Admissibility of the Likelihood Ratio Test......Page 419
9.10. Monotonicity of Power Functions of Tests of Independence of Sets......Page 420
9.11. Elliptically Contoured Distributions......Page 422
Problems......Page 426
10 Testing Hypotheses of Equality of Covariance Matrices and Equality of Mean Vectors and Covariance Matrices......Page 429
10.1. Introduction......Page 429
10.2. Criteria for Testing Equality of Several Covariance Matrices......Page 430
10.3. Criteria for Testing That Several Normal Distributions Are Identical......Page 433
10.4. Distributions of the Criteria......Page 435
10.5. Asymptotic Expansions of the Distributions of the Criteria......Page 442
10.6. The Case of Two Populations......Page 445
10.7. Testing the Hypothesis That a Covariance Matrix Is Proportional to a Given Matrix; The Sphericity Test......Page 449
10.8. Testing the Hypothesis That a Covariance Matrix Is Equal to a Given Matrix......Page 456
10.9. Testing the Hypothesis That a Mean Vector and a Covariance Matrix Are Equal to a Given Vector and Matrix......Page 462
10.10. Admissibility of Tests......Page 464
10.11. Elliptically Contoured Distributions......Page 467
Problems......Page 472
11 Principal Components......Page 477
11.1. Introduction......Page 477
11.2. Definition of Principal Components in the Population......Page 478
11.3. Maximum Likelihood Estimators of the Principal Components and Their Variances......Page 485
11.4. Computation of the Maximum Likelihood Estimates of the Principal Components......Page 487
11.5. An Example......Page 489
11.6. Statistical Inference......Page 491
11.7. Testing Hypotheses about the Characteristic Roots of a Covariance Matrix......Page 496
11.8. Elliptically Contoured Distributions......Page 500
Problems......Page 501
12 Canonical Correlations and Canonical Variables......Page 505
12.1. Introduction......Page 505
12.2. Canonical Correlations and Variates in the Population......Page 506
12.3. Estimation of Canonical Correlations and Variates......Page 516
12.4. Statistical Inference......Page 521
12.5. An Example......Page 523
12.6. Linearly Related Expected Values......Page 526
12.7. Reduced Rank Regression......Page 532
12.8. Simultaneous Equations Models......Page 533
Problems......Page 544
13 The Distributions of Characteristic Roots and Vectors......Page 546
13.1. Introduction......Page 546
13.2. The Case of Two Wishart Matrices......Page 547
13.3. The Case of One Nonsingular Wishart Matrix......Page 556
13.4. Canonical Correlations......Page 561
13.5. Asymptotic Distributions in the Case of One Wishart Matrix......Page 563
13.6. Asymptotic Distributions in the Case of Two Wishart Matrices......Page 567
13.7. Asymptotic Distribution in a Regression Model......Page 573
13.8. Elliptically Contoured Distributions......Page 581
Problems......Page 585
14 Factor Analysis......Page 587
14.1. Introduction......Page 587
14.2. The Model......Page 588
14.3. Maximum Likelihood Estimators for Random Orthogonal Factors......Page 594
14.4. Estimation for Fixed Factors......Page 604
14.5. Factor Interpretation and Transformation......Page 605
14.6. Estimation for Identification by Specified Zeros......Page 608
14.7. Estimation of Factor Scores......Page 609
Problems......Page 611
15 Patterns of Dependence; Graphical Models......Page 613
15.1. Introduction......Page 613
15.2. Undirected Graphs......Page 614
15.3. Directed Graphs......Page 622
15.4. Chain Graphs......Page 628
15.5. Statistical Inference......Page 631
Appendix A Matrix Theory......Page 642
A.1. Definition of a Matrix and Operations on Matrices......Page 642
A.2. Characteristic Roots and Vectors......Page 649
A.3. Partitioned Vectors and Matrices......Page 653
A.4. Some Miscellaneous Results......Page 657
A.5. Gram-Schmidt Orthogonalization and the Solution of Linear Equations......Page 665
Appendix B Tables......Page 669
B.1. Wilks' Likelihood Criterion: Factors C(p, m, M) to Adjust to X^2_{p,m}, where M = n-p+1......Page 669
B.2. Tables of Significance Points for the Lawley-Hotelling Trace Test......Page 675
B.3. Tables of Significance Points for the Bartlett-Nanda-Pillai Trace Test......Page 691
B.4. Tables of Significance Points for the Roy Maximum Root Test......Page 695
B.5. Significance Points for the Modified Likelihood Ratio Test of Equality of Covariance Matrices Based on Equal Sample Sizes......Page 699
B.6. Correction Factors for Significance Points for the Sphericity Test......Page 701
B.7. Significance Points for the Modified Likelihood Ratio Test \\Sigma = \\Sigma_0......Page 703
References......Page 705
Index......Page 731

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