Robust empirical Bayes tests for continuous distributions

In this paper, we study the empirical Bayes two-action problem under linear loss function. Upper bounds on the regret of empirical Bayes testing rules are investigated. Previous results on this problem construct empirical Bayes tests using kernel type estimators of nonparametric functionals. Further, they have assumed specific forms, such as the continuous one-parameter exponential family for {Fθ:θ∈Ω}, for the family of distributions of the observations. In this paper, we present a new general approach of establishing upper bounds (in terms of rate of convergence) of empirical Bayes tests for this problem. Our results are given for any family of continuous distributions and apply to empirical Bayes tests based on any type of nonparametric method of functional estimation. We show that our bounds are very sharp in the sense that they reduce to existing optimal or nearly optimal rates of convergence when applied to specific families of distributions.