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(Wiley Series in Probability and Statistics): Aspects of Multivariate Statistical Theory pdf 格式
Contents
TABLES xvii
COMMONLY USED NOTATION xix
I . THE MULTIVARIATE NORMAL AND RELATED DISTRIBUTIONS 1
I . I . Introduction, I
1.2. The Multivariate Normal Distribution, 2
I .2.1. Definition and Properties, 2
I .2.2. Asymptotic Distributions of Sample Means and Covariance
Matrices, 15
I .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. I. Jacobians, Exterior Products, and Related Topics, 50
2. I . 1. Jacobians and Exterior Products, 50
2. I .2. The Multivariate Gamma Function, 6 I
2. I .3. More Jacobians, 63
2. I .4. Invariant Measures, 67
2.2. Kronecker Products, 73
Problems, 76
xi
3. SAMPLES FROM A MULTIVARIATE NORMAL I)ISTRIBUTION,
AND THE WISHAIlT AND MULTIVARIATE BETA
DISTRIDUTIONS 79
3. I.
3.2. The Wishart Distribution, 85
Samples From a Multivariate Normal Distribution and Maximum
Likelihood Estimation of the Parameters, 79
3.2.1. The Wishart Density Function, 85
3.2.2. Characteristic Function, Moments, and Asymptotic
Distribution, 57
3.2.3. Some Properties of the Wishart Distribution, 91
3.2,4. Bartlett’s Dccomposition 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, 12 I
4.2. Estimation of the Mean, 122
4.3. Estimation of the Covariance Matrix, 128
4.4. Estimation of the Precision Matrix, 136
Problems, 14 I
5. CORRELATION COEFFICIENTS
5. I. Ordinary Correlation Coefficients, 144
5.1.1. Introduction, 144
5. I .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, I51
5.1.4. Asymptotic Distribution of a Sample Correlation
Coefficient From an Elliptical Distribution, 157
5. I .5. Testing Hypothcses about I’opulation Correlation
Coefficients, 160
5.2. The Multiple Correlation Coefficient, 164
5.2. I. Introduction, 164
5.2.2. Distribution of the Sample Multiple Correlation Coefficicnt
in the Case of Independcncc, 167
144
Conrenrs xiii
5.2.3. The Non-null Distribution of a Sample Multipfe Correlation
Coefficient in the Case of Normality, 171
5.2.4. Asymptotic Distributions of a Sample Multiple Correlation
Coefficient from an Elliptical Distribution, I79
5.2.5. Testing Hypotheses about a Population Multiple
Correlation Coefficient, 185
5.3. Partial Correlation Coefficients, 187
Problems, I89
6. INVARIANT TESTS AND SOME APPLICATIONS 196
6. I . Invariance and Invariant Tests, 196
6.2. The Multiple Correlation Coefficient and Invariance, 206
6.3. Hotelling’s T2 Statistic and Invariance, 21 I
Problems, 2 I9
7. ZONAL POLYNOMIALS AND SOME FUNCTIONS OF
MATRIX ARGUMENT 225
7. I . Introduction, 225
7.2. Zonal Polynomials, 227
7.2. I.
7.2.2. A Fundamental Property, 239
7.2.3. Some Basic Integrals, 246
7.3. Hypergeometric Functions 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, 28 I
Problems, 286
Definition and Construction, 227
8. SOME STANDARD TESTS ON COVARIANCE MATRICES AND
MEAN VECTORS 29 I
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, 3 I 1
8.2.6. Asymptotic Non-null Distributions of the Modificd
Likelihood Ratio Statistic when r = 2, 3 16
8.2.7. The Asymptotic Null Distribution of the Modificd
Likelihood Ratio Statistic for Elliptical Samples, 329
8.2.8. Other Test Statistics, 33 I
8.3. I . The Likelihood Ratio Statistic; Invariance and
Unbiasedness, 333
8.3.2. Momcnts of the Likelihood Ratio Statistic. 339
8.3.3. The Asymptotic Null Distribution of tlic 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
Testing That a Covariance Matrix Equals a Specified Matrix, 353
8.4. I . The Likelihood Ratio Test and Invariance, 353
8.4.3. Unbiascdness and the Modified Likelihood Ratio Test, 356
8.4.3. Mornents of the Modified Likelihood Ratio Statistic, 358
8.4.4. The Asymptotic Null Distribution of the Modificd
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
8.3. The Sphericity Test, 333
8.4.
Problems, 376
9. PRINCIPAL COMPONENTS AND RELATED TOPICS 380
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