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A classical mathematical treatment of the techniques, distributions, and inferences based on the multivariate normal distribution. Introduces 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. Discusses recent advances in multivariate analysis, including decision theory and robustness. Also includes tables of percentage points of many of the standard likelihood statistics used in multivariate statistical procedures.

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
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