Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X~f([short parallel]x-[theta][short parallel]2) and let [delta][pi](X) be the generalized Bayes estimator of [theta] with respect to a spherically symmetric prior, [pi]([short parallel][theta][short parallel]2), for loss [short parallel][delta]-[theta][short parallel]2. We show that if [pi](t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator [delta]0(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e-[alpha]t[beta] and e-[alpha]t+[beta][phi](t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .