Ensemble minimaxity of James-Stein estimators

TL;DR

This paper explores the ensemble minimaxity of James-Stein estimators for heteroscedastic multivariate normal means, showing they can be minimax with respect to ensemble risk, extending empirical Bayes insights.

Contribution

It introduces the concept of ensemble minimaxity for James-Stein estimators under heteroscedasticity, linking minimaxity to empirical Bayes methods.

Findings

James-Stein estimators can be ensemble minimax under heteroscedasticity

Shrinking more on coordinates with larger variances is beneficial

Ensemble minimaxity extends traditional minimax concepts

Abstract

This article discusses estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators shrinking more on the coordinates with larger variances, seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that such James-Stein type estimators can be ensemble minimax, minimax with respect to the ensemble risk, related to empirical Bayes perspective of Efron and Morris.