Empirical bayes tests in a positive exponential family with partial information on priors

Abstract

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(−x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C1, C2]. The value C1 is used in the construction of the proposed empirical Bayes test δ* n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C1, C2] δ* n is asymptotically optimal at a rate of convergence of order O(n−1/n n). Also, δ* n is robust in the sense that δ* n still possesses the asymptotic optimality even the assumption that "C1≦θ≦C2 may not hold.