Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X ∼ f ( ∥ x - θ ∥ 2 ) and let δ π ( X ) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π ( ∥ θ ∥ 2 ) , for loss ∥ δ - θ ∥ 2 . We show that if π ( t ) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ 0 ( X ) = X under certain conditions on f ( ) . The class of priors includes priors of the form 1 A + ∥ θ ∥ 2 k for k ⩽ p 2 - 1 and hence includes the fundamental harmonic prior 1 ∥ θ ∥ p - 2 . The class of sampling distributions includes certain variance mixtures of normals and other functions f ( t ) of the form e - α t β and e - α t + β φ ( 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 δ π ( X ) = 1 - ar ( t ) t X .