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Intro; Preface; Contents; 1 Decision Theory Preliminaries; 1.1 Introduction; 1.2 The Multivariate Normal Distribution; 1.3 The Uniform Distribution on a Sphere; 1.4 Bayesian Decision Theory; 1.5 Minimaxity; 1.6 Admissibility; 1.7 Invariance; 2 Estimation of a Normal Mean Vector I; 2.1 Introduction; 2.2 Some Intuition into Stein Estimation; 2.2.1 Best Linear Estimators; 2.2.2 Some Geometrical Insight; 2.2.3 The James-Stein Estimator as an Empirical Bayes Estimator; 2.3 Improved Estimators via Stein's Lemma; 2.4 James-Stein Estimators and Other Improved Estimators; 2.4.1 James-Stein Estimators
2.4.2 Positive-Part and Baranchik-Type Estimators2.4.3 Unknown Variance; 2.4.4 Estimators That Shrink Toward a Subspace; 2.5 A Link Between Stein's Lemma and Stokes' Theorem; 2.6 Differential Operators and Dimension Cut-Off When Estimating a Mean; 3 Estimation of a Normal Mean Vector II; 3.1 Bayes Minimax Estimators; 3.1.1 A Sufficient Condition for Minimaxity of (Proper, Generalized, and Pseudo) Bayes Estimators; 3.1.2 Construction of (Proper and Generalized) Minimax Bayes Estimators; 3.1.3 Examples; 3.1.4 Multiple Shrinkage Estimators; 3.2 Bayes Estimators in the Unknown Variance Case
3.5.3 Penalized Likelihood and Hierarchical Priors3.6 Estimation of a Predictive Density; 3.6.1 The Kullback-Leibler Divergence; 3.6.2 The Bayesian Predictive Density; 3.6.3 Sufficiency Reduction in the Normal Case; 3.6.4 Properties of the Best Invariant Density; 3.6.5 An Explicit Expression for U and Its Risk in the Normal Case; 4 Spherically Symmetric Distributions; 4.1 Introduction; 4.2 Spherically Symmetric Distributions; 4.3 Elliptically Symmetric Distributions; 4.4 Marginal and Conditional Distributions for Spherically Symmetric Distributions; 4.5 The General Linear Model
4.5.1 The Canonical Form of the General Linear Model4.5.2 Least Squares, Unbiased and Shrinkage Estimation; 4.5.3 Sufficiency in the General Linear Model; 4.5.4 Completeness for the General Linear Model; 4.6 Characterizations of the Normal Distribution; 5 Estimation of a Mean Vector for Spherically Symmetric Distributions I: Known Scale; 5.1 Introduction; 5.2 Baranchik-Type Estimators; 5.2.1 Variance Mixtures of Normal Distributions; 5.2.2 Densities with Tails Flatter Than the Normal; 5.3 More General Minimax Estimators; 5.4 Bayes Estimators; 5.5 Shrinkage Estimators for Concave Loss
2.4.2 Positive-Part and Baranchik-Type Estimators2.4.3 Unknown Variance; 2.4.4 Estimators That Shrink Toward a Subspace; 2.5 A Link Between Stein's Lemma and Stokes' Theorem; 2.6 Differential Operators and Dimension Cut-Off When Estimating a Mean; 3 Estimation of a Normal Mean Vector II; 3.1 Bayes Minimax Estimators; 3.1.1 A Sufficient Condition for Minimaxity of (Proper, Generalized, and Pseudo) Bayes Estimators; 3.1.2 Construction of (Proper and Generalized) Minimax Bayes Estimators; 3.1.3 Examples; 3.1.4 Multiple Shrinkage Estimators; 3.2 Bayes Estimators in the Unknown Variance Case
3.5.3 Penalized Likelihood and Hierarchical Priors3.6 Estimation of a Predictive Density; 3.6.1 The Kullback-Leibler Divergence; 3.6.2 The Bayesian Predictive Density; 3.6.3 Sufficiency Reduction in the Normal Case; 3.6.4 Properties of the Best Invariant Density; 3.6.5 An Explicit Expression for U and Its Risk in the Normal Case; 4 Spherically Symmetric Distributions; 4.1 Introduction; 4.2 Spherically Symmetric Distributions; 4.3 Elliptically Symmetric Distributions; 4.4 Marginal and Conditional Distributions for Spherically Symmetric Distributions; 4.5 The General Linear Model
4.5.1 The Canonical Form of the General Linear Model4.5.2 Least Squares, Unbiased and Shrinkage Estimation; 4.5.3 Sufficiency in the General Linear Model; 4.5.4 Completeness for the General Linear Model; 4.6 Characterizations of the Normal Distribution; 5 Estimation of a Mean Vector for Spherically Symmetric Distributions I: Known Scale; 5.1 Introduction; 5.2 Baranchik-Type Estimators; 5.2.1 Variance Mixtures of Normal Distributions; 5.2.2 Densities with Tails Flatter Than the Normal; 5.3 More General Minimax Estimators; 5.4 Bayes Estimators; 5.5 Shrinkage Estimators for Concave Loss