On solutions for global Stein optimization problems with applications

Abstract

Abstract This paper investigates generalized Stein estimation under spherically symmetric distributions. We reconsider important results on possible improvements over the best invariant estimator by shrinkage estimators through a successful analytic technique via a global optimization problem. Specifically, minimaxity of the James–Stein estimators for p-variate shift models with p⩾4 and general spherically symmetric distributions is instructively demonstrated. In addition, we derive a robustness result for the (formal) Bayes estimators of Stein (Ann. Statist. 9 (1981)) by showing that they continue to be minimax for certain non-normal distributions.