Proper Priors Yielding Linear Bayes Estimators for the Natural Parameter of an Exponential Family

Let X,X1, X2, . . . , Xn be i.i.d. random variables distributed according to an exponential family with natural parameter θ ∈ Ω. A family of priors, not necessarily proper, given in Ghosh and Meeden (1977), Ralescu and Ralescu (l981), and Brown (1986), gives normal posterior distributions and generates linear generalized Bayes estimators for θ when Ω is the real line. Here, necessary and sufficient conditions are given for the priors to be proper in the case that X has unbounded support, and when those conditions hold, the proper priors are shown to be a mixture of normal distributions. Admissibility of the resulting linear estimators is also proven. In the case that P (|X | ≤ A) = 1, for some A < ∞, the resulting (admissible) Bayes estimators have the curious property that a proper prior distribution may be chosen so as to assign arbitrarily small prior probability to the range of the estimator. In the case of unbounded support, the prior probability assigned to the range of the estimator is bounded below by 1/2.