On an empiricial Bayes test for a normal mean

We exhibit an empirical Bayes test delta(n)(*) for the normal mean testing problem using a linear error loss. Under the condition that the critical point of a Bayes test is within some known compact interval, delta(n)(*) is shown to be asymptotically optimal and its associated regret Bayes risk converges to zero at a rate O(n(-1)(ln n)(1.5)), where n is the number of past experiences available when the current component decision problem is considered. Under the same condition this rate is faster than the optimal rate of convergence claimed by Karunamuni.