Improving on the James-Stein Positive-Part Estimator

Abstract

The purpose of this paper is to give an explicit estimator dominating the positive-part James—Stein rule. The James—Stein estimator improves on the "usual" estimator X of a multivariate normal mean vector O if the dimension p of the problem is at least 3. It has been known since at least 1964 that the positive-part ver- Sion Of this estimator improves on the James—Stein estimator. Brown's 1971 results imply that the positive-part version is itself inadmissible although this result was assumed to be true much earlier. Explicit improvements, however, have not previously been found; indeed, 1988 results of Bock and of Brown imply that no estimator dominating the positive-part estimator exists whose unbiased estimator of risk is uniformly smaller than that of the positive-part estimator.