Identifiability of differentiable bayes estimators of the uniform scale parameter

Abstract

The problem of estimating the uniform scale parameter under the squared error loss function is investigated from a Bayesian viewpoint. A complete characterization of differentiable Bayes estimators and generalized Bayes estimators is given. The solution determines a family of prior measures both proper and improper, involving densities whose support is the whole parameter space, i.e, the interval (0,00)' Relations between degrees of smoothness of the estimators and the priors are investigated. We will also consider sequences, depending on the sample size, of Bayes (generalized Bayes) estimators with a fixed structure which are generated from a unique prior measure. They will be named strong Bayes sequences or strong generalized Bayes sequences. We characterize this type of Bayes estimation which is more restrictive than the usual one. As a consequence oithe characterization results, we will prove that strong Bayes sequences of polynomial form are not possible for the uniform scale parameter. Moreover we will show that the sequence whose components are the minimum risk equivariant estimator for each sample size is the best strong generalized Bayes sequence of polynomial form.