Sufficient Conditions for the Admissibility Under Squared Errors Loss of Formal Bayes Estimators

Abstract

This paper is concerned with the problem of finding reasonably explicit sufficient conditions for the almost admissibility of formal Bayes estimators (for definitions, see Section 2), where the underlying distribution is assumed known up to a single real parameter, and a real function of this parameter is being estimated with squared error as loss. The parameter space is assumed to be a possibly unbounded interval. These conditions are derived in Section 3. They are similar, in appearance, to results obtained by Karlin [3] when the underlying distribution is a member of the family of one parameter exponential distributions and the mean of this distribution is being estimated. The results of Section 3 should be viewed as a refinement of a heuristic argument given by Stein ([6] and [7] pages 233-240). In Section 2, some preliminary results and definitions are given. The results of Section 3 are applied in Section 4 to problems involving either the one dimensional exponential family or the estimation of a function of a single location parameter. Some of the results are known, in at least a similar form, while others are new. A counterexample based on a one dimensional location parameter problem is given in Section 5. It suggests that conditions of the type obtained here may even be necessary.