Stein Estimator from a Polar Coordinate Perspective

For simultaneously estimating three or more mean parameters from independent normal distributions with a 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 a closed-form expression. This astonishing discovery—better uniformly with explicit form when parameter dimension ≥ 3 has inspired a large amount of ongoing research. Statistical Science organized a special issue 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 the risk function of Stein's rule. This paper provides a third approach for solving this problem from a polar coordinate perspective. The new perspective is a useful complement in its own right, and meanwhile plays an important role for checking absolute integrability in the course of using Stein's lemma. Besides, there is relatively little work in the literature focusing specifically on heteroscedastic normal models that are as elegant as Stein's original work, with closed-form expressions and exact risk computations. 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. The findings in this paper provide a useful reference for further studies in this direction.