Stein Estimation for Spherically Symmetric Distributions: Recent Developments

TL;DR

This paper reviews recent advances in Stein-type shrinkage estimation for spherically symmetric distributions, emphasizing theoretical developments, robustness, and applications to constrained estimation problems.

Contribution

It generalizes the Stein lemma for spherically symmetric distributions and explores distributional robustness and minimax properties of estimators.

Findings

Generalized Stein lemma for spherically symmetric distributions

Development of robust, minimax estimators with residual vectors

Application to constrained location estimation in polyhedral cones

Abstract

This paper reviews advances in Stein-type shrinkage estimation for spherically symmetric distributions. Some emphasis is placed on developing intuition as to why shrinkage should work in location problems whether the underlying population is normal or not. Considerable attention is devoted to generalizing the "Stein lemma" which underlies much of the theoretical development of improved minimax estimation for spherically symmetric distributions. A main focus is on distributional robustness results in cases where a residual vector is available to estimate an unknown scale parameter, and, in particular, in finding estimators which are simultaneously generalized Bayes and minimax over large classes of spherically symmetric distributions. Some attention is also given to the problem of estimating a location vector restricted to lie in a polyhedral cone.