Universal Bayes Estimators

Abstract

Let $x_1, \cdots, x_n$ be i.i.d. random variables with a distribution depending on the real parameter. Under what conditions is a generalized Bayes estimator independent of the choice of the even loss function? The known answer to this question is that this independence holds if the posterior density is symmetric and unimodal. The description of distributions and corresponding generalized prior densities on the real line, for which the posterior density is symmetric and unimodal, is presented. These families form an important subclass of all exponential laws with two-dimensional sufficient statistics.