Robust generalized Bayes minimax estimators of location vectors for spherically symmetric distributions with unknown scale

Abstract

We consider estimation of the mean vector, θ, of a spherically symmetric distribution with unknown scale parameter σ under scaled quadratic loss. We show minimaxity of generalized Bayes estimators corresponding to priors of the form π(‖θ‖²)η^b where η = 1 / σ², for π(⋅) superharmonic with a non decreasing Laplacian under conditions on b and weak moment conditions. Furthermore, these generalized Bayes estimators are independent of the underlying density and thus have the strong robustness property of being simultaneously generalized Bayes and minimax for the entire class of spherically symmetric distributions.