Springer, 2018. — 339 p. — (Springer Series in Statistics). — ISBN: 303002184X.
This book provides a coherent framework for understanding shrinkage estimation in statistics. The term refers to modifying a classical estimator by moving it closer to a target which could be known a priori or arise from a model. The goal is to construct estimators with improved statistical properties. The book focuses primarily on point and loss estimation of the mean vector of multivariate normal and spherically symmetric distributions. Chapter 1 reviews the statistical and decision theoretic terminology and results that will be used throughout the book.
Chapter 2 is concerned with estimating the mean vector of a multivariate normal distribution under quadratic loss from a frequentist perspective. In Chapter 3 the authors take a Bayesian view of shrinkage estimation in the normal setting. Chapter 4 introduces the general classes of spherically and elliptically symmetric distributions. Point and loss estimation for these broad classes are studied in subsequent chapters. In particular, Chapter 5 extends many of the results from Chapters 2 and 3 to spherically and elliptically symmetric distributions. Chapter 6 considers the general linear model with spherically symmetric error distributions when a residual vector is available. Chapter 7 then considers the problem of estimating a location vector which is constrained to lie in a convex set. Much of the chapter is devoted to one of two types of constraint sets, balls and polyhedral cones. In Chapter 8 the authors focus on loss estimation and data-dependent evidence reports. Appendices cover a number of technical topics including weakly differentiable functions; examples where Stein's identity doesn't hold; Stein's lemma and Stokes' theorem for smooth boundaries; harmonic, superharmonic and subharmonic functions; and modified Bessel functions.
Decision Theory Preliminaries
Estimation of a Normal Mean Vector I
Estimation of a Normal Mean Vector II
Spherically Symmetric Distributions
Estimation of a Mean Vector for Spherically Symmetric Distributions I: Known Scale
Estimation of a Mean Vector for Spherically Symmetric Distributions II:With a Residual
Restricted Parameter Spaces
Loss and Confidence Level Estimation
AppendixWeakly Differentiable Functions
Examples of Weakly Differentiable Functions
Vanishing of the Bracketed Term in Stein’s Identity
Examples of Settings Where Stein’s Identity Does Not Hold
Stein’s Lemma and Stokes’ Theorem for Smooth Boundaries
Proof of Lemma
An Expression of the Haff Operator
Harmonic, Superharmonic and Subharmonic Functions
Differentiation of Marginal Densities
Results on Expectations and Integrals
Modified Bessel Functions