A Tribute to Charles Stein

TLDR: In this paper, a 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.

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. The following ten papers assembled in this volume represent some of the many areas into which shrinkage has expanded (a one-dimensional pun, no doubt). Clearly, the shrinkage literature has branched out substantially since 1956, the many contributors and the breadth of theory and practice being now far too large to cover with any degree of completeness in a review issue such as this one. But what these papers do show is the lasting impact of Stein (1956), and the ongoing vitality of the huge area that he catalyzed.