Generalized Multivariate Analysis

The theory of generalized multivariate analysis, based on elliptically contoured distributions, represents a brilliant achievement in the field of multivariate analysis. This is the first book on the subject. The text discusses estimation of parameters, testing of hypotheses, and linear models employing the method of stochastic representation, rather than following the classical treatments. It is designed as a textbook for a one-semester course at postgraduate level and as a reference source for lecturers and researchers. Cover Title page Preface CHAPTER I SOME MATRIX THEORY AND INVARIANCE 1.1. Definitions 1.1.1. Matrices 1.1.2. Determinants 1.1.3. Inverse of a Matrix 1.1.4. Partition of Matrices 1.1.5. Rank of a Matrix 1.1.6. Trace of a Matrix 1.1.7. Characteristic Roots and Characteristic Vectors 1.1.8. Positive Definite Matrices 1.1.9. Projection Matrices 1.2. Some Matrix Factorizations 1.3. Generalized Inverse of Matrix 1.4. "vec” Operator and Kronecker Products 1.4.1. "vec” Operator 1.4.2. Kronecker Products 1.4.3. Permutation Matrix 1.5. Derivatives of Matrices and the Matrix Differential 1.5.1. The Derivatives of Matrices with Respect to a Scalar 1.5.2. The Derivative of Scalar Functions of a Matrix with Respect to the Matrix 1.5.3. The Derivatives of Vectors 1.5.4. The Matrix Differential 1.6. Evaluation of the Jacobians of Some Transformations 1.7. Groups and Invariance References Exercises 1 CHAPTER II ELLIPTICALLY CONTOURED DISTRIBUTIONS 2.1. Multivariate Distributions 2.1.1. Multivariate Cumulative Distribution Function 2.1.2. Density 2.1.3. Marginal Distributions 2.1.4. Conditional Distributions 2.1.5. Independence 2.1.6. Characteristic Functions 2.1.7. The Operation 2.2. Moments of Multivariate Distributions 2.3. Multivariate Normal Distribution 2.4. Dirichlet Distribution 2.5. Spherical Distributions 2.5.1. Uniform Distribution and Its Stochastic Representation 2.5.2. Densities 2.5.3. The Class Phi_∞ 2.5.4. Invariant Distribution 2.6. Elliptically Contoured Distributions 2.6.1. The Stochastic Representation 2.6.2. Combination and Marginal Distributions 2.6.3. Moments 2.6.4. Conditional Distributions 2.6.5. Densities 2.7. Characterizations of Normality 2.8. Distributions of Quadratic Forms and Cochran’s Theorems 2.8.1. Distributions of Quadratic Forms 2.8.2. Cochran’s Theorem for the Normal Case 2.8.3. Cochran’s Theorem for the Case of ECD 2.9. Some Non-Central Distributions 2.9.1. Generalized Non-Central χ2—Distribution 2.9.2. Generalized Non-Central t—Distribution 2.9.3. Generalized Non-Central F—Distribution References Exercises 2 CHAPTER III SPHERICAL MATRIX DISTRIBUTIONS 3.1. Introduction 3.1.1. Left—Spherical Distributions 3.1.2. Spherical distributions 3.1.3. Multivariate Spherical Distributions 3.1.4. Vector—Spherical Distributions 3.2. Relationships among Classes of Spherical Matrix Distributions .97 3.2.1. Inclusion Relation 3.2.2. Classes of Marginal Distributions 3.2.3. Coordinate Systems 3.2.4. Densities 3.3. Elliptically Contoured Matrix Distributions 3.4. Distributions of Quadratic Forms 3.4.1. Densities of W 3.4.2. A Multivariate Analogue to Cochran’s Theorem 3.5. Some Related Distributions with Spherical Matrix Distributions 3.5.1. The Matrix Variate Beta Distributions 3.5.2. The Matrix Variate Dirichlet Distributions 3.5.3. The Matrix Variate t—Distributions 3.5.4. The Matrix Variate F—Distributions 3.5.5. Some Inverted Matrix Variate Distributions 3.5.6. Some Distributions of the Characteristic Roots of Matrix Variate 3.6. The Generalized Bartlett Decomposition and the Spectral Decomposition of Spherical Matrix Distributions 3.6.1. Coordinate Transformations 3.6.2. The Generalized Bartlett Decomposition 3.6.3. The Spectral Decomposition References Exercises 3 CHAPTER IV ESTIMATION OF PARAMETERS 4.1. MLE’s of Mean Vector and Covariance Matrix 4.1.1. MLE’s of μ and Σ in VS_{x p}(μ,Σ,f) 4.1.2. Examples 4.1.3. MLE’s of μ and Σ in LS_{x p}(μ,Σ,f) and SS_{x p}(μ,Σ,f) 4.1.4. MLE’s of Parametric Functions 4.2. The Distributions of Some Estimators 4.2.1. Joint Density 4.2.2. Marginal Density 4.2.3. Independence of μ^{_} and S 4.2.4. Distribution of Sample Correlation Coefficients 4.3. Properties of ^μ and ^Σ 4.3.1. Unbiasedness 4.3.2. Sufficiency 4.3.3. Completeness 4.3.4. Consistency 4.4. Minimax and Admissible Characters of ^μ and Σ 4.4.1. Inadmissibffity of x^{_} as an Estimate of μ 4.4.2. Discussion on Estimation of 4.4.3. Minimax Estimates of t References Exercises 4 CHAPTER V TESTING HYPOTHESES 5.1. Distrbution-Free Statistics 5.2. Testing Hypotheses About Mean Vectors 5.2.1. Likelihood Ratio Criteria 5.2.2. Testing that a Mean Vector Equals a Specffied Vector 5.2.3. The Distribution of T2 5.2.4. T2-Testing and Invariance of Tests 5.2.5. Testing Equality of Several Means with Equal and Unknown Covariance Matrices 5.3. Tests for Covariance Matrices 5.3.1. The Spherical Test 5.3.2. Equality of Several Covariance Matrices 5.3.3. Simultaneously Testing Equality of Several Means and Covariance matrices 5.3.4. Testing Lack of Correlation Between Sets of Variates 5.4. A Note on Likelihood Ratio Method 5.5. Robust Tests with Invariance 5.5.1. Robust Tests for Spherical Symmetry 5.5.2. A Multivariate Test 5.6 Goodness of Fit Test for Elliptical Symmetry 5.6.1. A Characteristic of Spherical Symmetry 5.6.2. Significance Tests for Spherical Symmetry (I) 5.6.3. Signfficance Tests for Spherical Symmetry (II) 5.6.4. Significance Tests for Elliptical Symmetry References Exercises 5 CHAPTER VI LINEAR MODELS 6.1. Definition and Examples 6.1.1. Definition 6.1.2. Regression Model 6.1.3. Variance Analysis Model 6.1.4. Discriminant Analysis 6.2. BLUE 6.2.1. Least Squares Estimates 6.2.2. BLUE 6.2.3. Regularity 6.2.4. Variation of Models 6.3. Variance Components 6.3.1. Least Squares Method 6.3.2. Invariant QUE (IQUE) 6.3.3. MINQUE 6.4. Hypothesis Testing 6.4.1. Linear Hypothesis 6.4.2. Canonical Form 6.4.3. Pre-test Estimates and James—Stein Estimates 6.5. Applications 6.5.1. Double Screening Stepwise Regression (DSSR Method) 6.5.2. Example 6.5.3. Discriminant Analysis and Regression References Exercises 6 REFERENCES INDEX A study of multivariate analysis, discussing estimation of parameters, testing of hypotheses, and linear models employing the method of stochastic representation. This book is designed as a textbook for postgraduates and as a reference source for lecturers and researchers.