Characterization of Priors in the Stein Problem

Abstract

The so-called Stein problem is addressed in the estimation of a mean vector of a multivariate normal distribution with a known covariance matrix. For general prior distributions with sphericity, the paper derives conditions on priors under which the resulting generalized Bayes estimators are minimax relative to the usual quadratic loss. It is also shown that the conditions can be expressed based on the inverse Laplace transform of the general prior. Stein's super-harmonic condition is derived from the general conditions. Finally, the priors are characterized for the admissibility.