Bayes minimax estimators of a location vector for densities in the Berger class

Abstract

We consider Bayesian estimation of the location parameter θ of a random vector X having a unimodal spherically symmetric density f(‖x−θ‖2) when the prior density π(‖θ‖2) is spherically symmetric and superharmonic. We study minimaxity of the generalized Bayes estimator δπ(X)=X+∇M(X)/m(X) under quadratic loss, where m is the marginal associated to f(‖x−θ‖2) and M is the marginal with respect to F(‖x−θ‖2)=1/2∫‖x−θ‖2∞f(t) dt under the condition inf t≥0F(t)/f(t)=c>0 (see Berger [1]). We adopt a common approach to the cases where F(t)/f(t) is nonincreasing or nondecreasing and, although details differ in the two settings, this paper complements the article by Fourdrinier and Strawderman [7] who dealt with only the case where F(t)/f(t) is nondecreasing. When F(t)/f(t) is nonincreasing, we show that the Bayes estimator is minimax provided a ‖∇π(‖θ‖2)‖2/π(‖θ‖2)+2 c2 Δπ(‖θ‖2)≤0 where a is a constant depending on the sampling density. When F(t)/f(t) is nondecreasing, the first term of that inequality is replaced by b g(‖θ‖2) where b also depends on f and where g(‖θ‖2) is a superharmonic upper bound of ‖∇π(‖θ‖2)‖2/π(‖θ‖2). Examples illustrate the theory.