On empirical Bayes tests in a positive exponential family

We study empirical Bayes tests for testing the hypotheses H-0: theta less than or equal to theta(0) against H-1: theta > theta(0) in a positive exponential family having probability density u(x)c(theta)exp(-theta x), theta > 0, using a linear error loss. Under the assumption that the critical point a(G) of a Bayes test is within some known compact interval [C-1, C-2], where 0 < C-1 < C-2 < infinity, we are able to construct an empirical Bayes test delta(n)* 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. (C) 2000 Elsevier Science B.V. All rights reserved. MSC: 62C12.