From Minimax Shrinkage Estimation to Minimax Shrinkage Prediction

TL;DR

This paper reviews the development of minimax shrinkage estimators for multivariate normal means and predictive densities, highlighting theoretical advances and new estimator classes under various models and risks.

Contribution

It synthesizes historical and recent progress, emphasizing superharmonic conditions and introducing new minimax shrinkage estimators for different statistical settings.

Findings

Superharmonic conditions for minimaxity derived

Introduction of multiple shrinkage estimators

Extension to nonparametric regression models

Abstract

In a remarkable series of papers beginning in 1956, Charles Stein set the stage for the future development of minimax shrinkage estimators of a multivariate normal mean under quadratic loss. More recently, parallel developments have seen the emergence of minimax shrinkage estimators of multivariate normal predictive densities under Kullback--Leibler risk. We here describe these parallels emphasizing the focus on Bayes procedures and the derivation of the superharmonic conditions for minimaxity as well as further developments of new minimax shrinkage predictive density estimators including multiple shrinkage estimators, empirical Bayes estimators, normal linear model regression estimators and nonparametric regression estimators.