Shrinkage estimation for mean and covariance matrices. (2020)

• Record Type: Book
• Further Information: Note: Hisayuki Tsukuma, Tatsuya Kubokawa.
• Other Names: Tsukuma, Hisayuki; Kubokawa, Tatsuya, 1959-
• Contents: Intro -- 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
• Publisher Details: Singapore : Springer
• Publication Date: 2020
• Extent: 1 online resource (119 p.)
• Subjects: 519.5/42; Statistical decision; Statistical decision; Electronic books; Electronic books
• Languages: English
• ISBNs: 9789811515965; 9811515964
• Related ISBNs: 9811515956; 9789811515958
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.