Aspects of multivariate statistical theory /

TABLES xvii
COMMONLY USED NOTATION xix
I. THE MULTIVARIATE NORMAL AND RELATED DISTRIBUTIONS
1.1. Introduction, I
1.2. The Multivariate Normal Distribution, 2
1.2.1. Definition and Properties, 2
1.2.2. Asymptotic Distributions of Sample Means and Covariance
Matrices, 15
1.3. The Noncentral X2 and F Distributions, 20
1.4. Some Results on Quadratic Forms, 26
1.5. Spherical and Elliptical Distributions, 32
1.6. Multivariate Cumulants, 40
Problems, 42
2. JACOBIANS, EXTERIOR PRODUCTS, KRONECKER PRODUCTS,
AND RELATED TOPICS 50
2.1. Jacobians, Exterior Products, and Related Topics, 50
2.1.1. Jacobians and Exterior Products, 50
2.1.2. The Multivariate Gamma Function, 61
2.1.3. More Jacobians, 63
2.1.4. Invariant Measures, 67
2.2. Kronecker Products, 73
Problems, 76
3. SAMPLES FROM A MULTIVARIATE NORMAL DISTRIBUTION,
AND THE WISHART AND MULTIVARIATE BETA
DISTRIBUTIONS 79
3.1. Samples From a Multivariate Normal Distribution and Maximum
Likelihood Estimation of the Parameters, 79
3.2. The Wishart Distribution, 85
3.2.1. The Wishart Density Function, 85
3.2.2. Characteristic Function, Moments, and Asymptotic
Distribution, 87
3.2.3. Some Properties of the Wishart Distribution, 91
3.2.4. Bartlett's Decomposition and the Generalized Variance,
99
3.2.5. The Latent Roots of a Wishart Matrix, 103
3.3. The Multivariate Beta Distribution, 108
Problems, 112
4. SOME RESULTS CONCERNING DECISION-THEORETIC
ESTIMATION OF THE PARAMETERS OF A MULTIVARIATE
NORMAL DISTRIBUTION 121
4.1. Introduction, 121
4.2. Estimation of the Mean, 122
4.3. Estimation of the Covariance Matrix, 128
4.4. Estimation of the Precision Matrix, 136
Problems, 141
5. CORRELATION COEFFICIENTS 144
5.1. Ordinary Correlation Coefficients, 144
5.1.1. Introduction, 144
5.1.2. Joint and Marginal Distributions of Sample Correlation
Coefficients in the Case of Independence, 145
5.1.3. The Non-null Distribution of a Sample Correlation
Coefficient in the Case of Normality, 151
5.1.4. Asymptotic Distribution of a Sample Correlation
Coefficient From an Elliptical Distribution, 157
5.1.5. Testing Hypotheses about Population Correlation
Coefficients, 160
5.2. The Multiple Correlation Coefficient, 164
5.2.1. Introduction, 164
5.2.2. Distribution of the Sample Multiple Correlation Coefficient
in the Case of Independence, 167
5.2.3. The Non-null Distribution of a Sample Multiple Correlation
Coefficient in the Case of Normality, 171
5.2.4. Asymptotic Distributions of a Sample Multiple Correlation
Coefficient from an Elliptical Distribution, 179
5.2.5. Testing Hypotheses about a Population Multiple
Correlation Coefficient, 185
5.3. Partial Correlation Coefficients, 187
Problems, 189
6. INVARIANT TESTS AND SOME APPLICATIONS 196
6.1. Invariance and Invariant Tests, 196
6.2. The Multiple Correlation Coefficient and Invariance, 206
6.3. Hotelling's T2 Statistic and Invariance, 211
Problems, 219
7. ZONAL POLYNOMIALS AND SOME FUNCTIONS OF
MATRIX ARGUMENT 225
7.1. Introduction, 225
7.2. Zonal Polynomials, 227
7.2.1. Definition and Construction, 227
7.2.2. A Fundamental Property, 239
7.2.3. Some Basic Integrals, 246
7.3. Hypergeometric Functionk of Matrix Argument, 258
7.4. Some Results on Special Hypergeometric Functions, 262
7.5. Partial Differential Equations for Hypergeometric Functions, 266
7.6. Generalized Laguerre Polynomials, 281
Problems, 286
8. SOME STANDARD TESTS ON COVARIANCE MATRICES AND
MEAN VECTORS 291
8.1. Introduction, 291
8.2. Testing Equality of r Covariance Matrices, 291
8.2.1. The Likelihood Ratio Statistic and Invariance, 291
8.2.2. Unbiasedness and the Modified Likelihood Ratio Test, 296
8.2.3. Central Moments of the Modified Likelihood Ratio
Statistic, 301
8.2.4. The Asymptotic Null Distribution of the Modified
Likelihood Ratio Statistic, 303
8.2.5. Noncentral Moments of the Modified Likelihood Ratio
Statistic when r 2, 311
8.2.6. Asymptotic Non-null Distributions of the Modified
Likelihood Ratio Statistic when r = 2, 316
8.2.7. The Asymptotic Null Distribution of the Modified
Likelihood Ratio Statistic for Elliptical Samples, 329
8.2.8. Other Test Statistics, 331
8.3. The Sphericity Test, 333
8.3.1. The Likelihood Ratio Statistic; Invariance and
Unbiasedness, 333
8.3.2. Moments of the Likelihood Ratio Statistic, 339
8.3.3. The Asymptotic Null Distribution of the Likelihood Ratio
Statistic, 343
8.3.4. Asymptotic Non-null Distributions of the Likelihood Ratio
Statistic, 344
8.3.5. The Asymptotic Null Distribution of the Likelihood Ratio
Statistic for an Elliptical Sample, 351
8.3.6. Other Test Statistics, 353
8.4. Testing That a Covariance Matrix Equals a Specified Matrix, 353
8.4.1. The Likelihood Ratio Test and Invariance, 353
8.4.2. Unbiasedness and the Modified Likelihood Ratio Test, 356
8.4.3. Moments of the Modified Likelihood Ratio Statistic, 358
8.4.4. The Asymptotic Null Distribution of the Modified
Likelihood Ratio Statistic, 359
8.4.5. Asymptotic Non-null Distributions of the Modified
Likelihood Ratio Statistic, 362
8.4.6. The Asymptotic Null Distribution of the Modified
Likelihood Ratio Statistic for an Elliptical Sample, 364
8.4.7. Other Test Statistics, 365
8.5. Testing Specified Values for the Mean Vector and Covariance
Matrix, 366
8.5.1. The Likelihood Ratio Test, 366
8.5.2. Moments of the Likelihood Ratio Statistic, 369
8.5.3. The Asymptotic Null Distribution of the Likelihood Ratio
Statistic, 370
8.5.4. Asymptotic Non-null Distributions of the Likelihood Ratio
Statistic, 373
Problems, 376
9. PRINCIPAL COMPONENTS AND RELATED TOPICS 380
9.1. Introduction, 380
9.2. Population Principal Components, 381
9.3. Sample Principal Components, 384
9.4. The Joint Distribution of the Latent Roots of a Sample Covariance
Matrix, 388
9.5. Asymptotic Distributions of the Latent Roots of a Sample
Covariance Matrix, 390
9.6. Some Inference Problems in Principal Components, 405
9.7. Distributions of the Extreme Latent Roots of a Sample Covariance
Matrix, 420
Problems, 426
10. THE MULTIVARIATE LINEAR MODEL 429
10.1. Introduction, 429
10.2. A General Testing Problem: Canonical Form, Invariance, and
the Likelihood Ratio Test, 432
10.3. The Noncentral Wishart Distribution, 441
10.4. Joint Distributions of Latent Roots in MANOVA, 449
10.5. Distributional Results for the Likelihood Ratio Statistic, 455
10.5.1. Moments, 455
10.5.2. Null Distribution, 457
10.5.3. The Asymptotic Null Distribution, 458
10.5.4. Asymptotic Non-null Distributions, 460
10.6 Other Test Statistics, 465
10.6. I. Introduction, 465
10.6.2. The To2 Statistic, 466
10.6.3. The V Statistic, 479
10.6.4. The Largest Root, 481
10.6.5. Power Comparisons, 484
10.7. The Single Classification Model, 485
10.7.1. Introduction, 485
10.7.2. Multiple Discriminant Analysis, 488
10.7.3. Asymptotic Distributions of Latent Roots in MANOVA,
492
10.7.4. Determining the Number of Useful Discriminant
Functions, 499
10.7.5. Discrimination Between Two Groups, 504
10.8. Testing Equality of p Normal Populations, 507
10.8.1. The Likelihood Ratio Statistic and Moments, 507
10.8.2. The Asymptotic Null Distribution of the Likelihood
Ratio Statistic, 512
10.8.3. An Asymptotic Non-null Distribution of the Likelihood
Ratio Statistic, 513
Problems, 517
I I. TESTING INDEPENDENCE BETWEEN k SETS OF VARIABLES
AND CANONICAL CORRELATION ANALYSIS 526
11.1. Introduction, 526
11.2. Testing Independence of k Sets of Variables, 526
11.2.1. The Likelihood Ratio Statistic and Invariance, 526
11.2.2. Central Moments of the Likelihood Ratio Statistic, 532
11.2.3. The Null Distribution of the Likelihood Ratio
Statistic, 533
11.2.4. The Asymptotic Null Distribution of the Likelihood
Ratio Statistic, 534
11.2.5. Noncentral Moments of the Likelihood Ratio Statistic
when k = 2, 536
11.2.6 Asymptotic Non-null Distributions of the Likelihood
Ratio Statistic when k = 2, 542
11.2.7. The Asymptotic Null Distribution of the Likelihood
Ratio Statistic for Elliptical Samples, 546
11.28. Other Test Statistics, 548
11.3. Canonical Correlation Analysis, 548
11.3.1. Introduction, 548
11.3.2. Population Canonical Correlation Coefficients and
Canonical Variables, 549
11.3.3. Sample Canonical Correlation Coefficients and
Canonical Variables, 555
11.3.4. Distributions of the Sample Canonical Correlation
Coefficients, 557
11.3.5. Asymptotic Distributions of the Sample Canonical
Correlation Coefficients, 562
11.3.6. Determining the Number of Useful Canonical
Variables, 567
Problems, 569
APPENDIX. SOME MATRIX THEORY 572
Al. Introduction, 572
A2. Definitions, 572
A3. Determinants, 575
A4. Minors and Cofactors, 579
AS. Inverse of a Matrix, 579
A6. Rank of a Matrix, 582
A7. Latent Roots and Latent Vectors, 582
A8. Positive Definite Matrices, 585
A9. Some Matrix Factorizations, 586
BIBLIOGRAPHY 650
INDEX 663