A Tribute to Charles Stein

TL;DR

This paper commemorates Charles Stein's groundbreaking 1956 discovery that challenged traditional estimators in high-dimensional normal mean estimation, sparking extensive research on minimax shrinkage methods and Stein's influential contributions.

Contribution

It reviews developments in minimax shrinkage estimation inspired by Stein's work, highlighting the evolution of estimators that outperform traditional methods in high dimensions.

Findings

Stein's estimator dominates in dimensions 3 and higher.

Stein's Lemma provides a key technical tool for risk assessment.

The literature has expanded significantly since Stein's original discovery.

Abstract

In 1956, Charles Stein published an article that was to forever change the statistical approach to high-dimensional estimation. His stunning discovery that the usual estimator of the normal mean vector could be dominated in dimensions 3 and higher amazed many at the time, and became the catalyst for a vast and rich literature of substantial importance to statistical theory and practice. As a tribute to Charles Stein, this special issue on minimax shrinkage estimation is devoted to developments that ultimately arose from Stein's investigations into improving on the UMVUE of a multivariate normal mean vector. Of course, much of the early literature on the subject was due to Stein himself, including a key technical lemma commonly referred to as Stein's Lemma, which leads to an unbiased estimator of the risk of an almost arbitrary estimator of the mean vector.