Empirical Bayes Tests Based on Kernel Sequence Estimation

In this paper, we consider the hypothesis-testing problem in the continuous one-parameter exponential family using the nonparametric empirical Bayes approach. In order to estimate an unknown marginal density and its derivative, a kernel sequence method is introduced. This method uses a sequence of kernel functions and allows the kernel index and window bandwidth to vary simultaneously. Thus improved estimates are obtained. Then we construct a monotone empirical Bayes test based on these estimates and show that the rule has a rate of convergence of (lnn) /n for any > 0. This rate substantially improves the previous results and is much closer to the lower bound rate 1/n. Since the rule is monotone, it also has good performance for small samples.