Stein’s estimator from a polar-coordinate’s perspective

For simultaneously estimating three or larger mean parameters from independent normal distribution with common variance, Stein¹ proved the inadmissibility of the usual estimator, and constructed jointly with James a uniformly better estimator in the sense of mean squared error loss with closed-form expression. This astonishing discovery—better uniformly with explicit form when parameter dimension ≥ 3, inspires large amount of continuing research. Statistical Science organized a special section in 2012, “MINIMAX SHRINKAGE ESTIMATION: A TRIBUTE TO CHARLES STEIN”, to express continuing tribute to Charles Stein. The carefully designed transformation of James and Stein² and Stein’s basic lemma^3-4 are two basic approaches to computing risk function of Stein’s rule. This paper provides a third way for solving this problem from a polar-coordinate’s perspective. The new perspective is a useful complement on its own right, and meanwhile plays important role for checking absolute integrability in the course of using Stein’s lemma. Besides, there are relatively few work in the literature focusing particularly on heteroscedastic normal models that are as elegant as Stein’s original work, closed-form expression as well as exact risk computation. To this end, we provide a class of estimators with explicit structure inspired by James and Stein’s original construct, and find the most appropriate values of coeffcients among this class by direct computing and matching. Findings in this paper provide a useful reference for further studies in this direction.