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

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.