Shrinkage Estimation for Mean and Covariance Matrices (SpringerBriefs in Statistics)

Preface
Contents
1 Decision-Theoretic Approach to Estimation
	1.1 Decision-Theoretic Framework for Estimation
	1.2 James-Stein\'s Shrinkage Estimator
	1.3 Unbiased Risk Estimate and Stein\'s Identity
	References
2 Matrix Algebra
	2.1 Notation
	2.2 Nonsingular Matrix and the Moore-Penrose Inverse
	2.3 Kronecker Product and Vec Operator
	2.4 Matrix Decompositions
	References
3 Matrix-Variate Distributions
	3.1 Preliminaries
		3.1.1 The Multivariate Normal Distribution
		3.1.2 Jacobians of Matrix Transformations
		3.1.3 The Multivariate Gamma Function
	3.2 The Matrix-Variate Normal Distribution
	3.3 The Wishart Distribution
	3.4 The Cholesky Decomposition of the Wishart Matrix
	References
4 Multivariate Linear Model and Group Invariance
	4.1 Multivariate Linear Model
	4.2 A Canonical Form
	4.3 Group Invariance
	References
5 A Generalized Stein Identity and Matrix Differential Operators
	5.1 Stein\'s Identity in Matrix-Variate Normal Distribution
	5.2 Some Useful Results on Matrix Differential Operators
	References
6 Estimation of the Mean Matrix
	6.1 Introduction
	6.2 The Unified Efron-Morris Type Estimators Including Singular Cases
		6.2.1 Empirical Bayes Methods
		6.2.2 The Unified Efron-Morris Type Estimator
	6.3 A Unified Class of Matricial Shrinkage Estimators
	6.4 Unbiased Risk Estimate
	6.5 Examples for Specific Estimators
		6.5.1 The Unified Efron-Morris Type Estimator
		6.5.2 A Modified Stein-Type Estimator
		6.5.3 Modified Efron-Morris Type Estimator
	6.6 Related Topics
		6.6.1 Positive-Part Rule Estimators
		6.6.2 Shrinkage Estimation with a Loss Matrix
		6.6.3 Application to a GMANOVA Model
		6.6.4 Generalization in an Elliptically Contoured Model
	References
7 Estimation of the Covariance Matrix
	7.1 Introduction
	7.2 Scale Invariant Estimators
	7.3 Triangular Invariant Estimators and the James-Stein Estimator
		7.3.1 The James-Stein Estimator
		7.3.2 Improvement Using a Subgroup Invariance
	7.4 Orthogonally Invariant Estimators
		7.4.1 Class of Orthogonally Invariant Estimators
		7.4.2 Unbiased Risk Estimate
		7.4.3 Examples
	7.5 Improvement Using Information on Mean Statistic
		7.5.1 A Class of Estimators and Its Risk Function
		7.5.2 Examples of Improved Estimators
		7.5.3 Further Improvements with a Truncation Rule
	7.6 Related Topics
		7.6.1 Decomposition of the Estimation Problem
		7.6.2 Decision-Theoretic Studies Under Quadratic Losses
		7.6.3 Estimation of the Generalized Variance
		7.6.4 Estimation of the Precision Matrix
	References
Index