Selection of the best Bernoulli population provided it is better than a control: An empirical Bayes approach

We investigate an empirical Bayes procedure δn* for selecting the best Bernoulli population from among k competitors provided it is better than a control based on negative binomial sampling. The performance of δn* is evaluated in terms of its associated Bayes risk. Under some conditions, it is shown that δn* is asymptotically optimal, and the associated Bayes risk converges to the minimum Bayes risk at an exponential decay rate exp(-cn) for some positive number c, where n is the number of historical data at hand when the present selection problem is considered. A simulation is also carried out to study the performance of δn* for small to moderate values of n.