Generalized Multivariate Analysis

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