On empirical Bayes tests in a positive exponential family

Abstract

We study empirical Bayes tests for testing the hypotheses H0:θ⩽θ0 against H1:θ>θ0 in a positive exponential family having probability density u(x)c(θ)exp(−θx),θ>0, using a linear error loss. Under the assumption that the critical point aG of a Bayes test is within some known compact interval [C1,C2], where 0<C1<C2<∞, we are able to construct an empirical Bayes test δn∗ possessing asymptotic optimality, with regret converging to zero at a rate of order O(n−s/(s+3)), where s is an arbitrary positive integer. This rate of convergence has improved the earlier existing rate of convergence of empirical Bayes tests regarding the underlying testing problem in the literature.