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Robust empirical Bayes tests for discrete distributions

In this paper, we investigate the empirical Bayes (EB) linear loss two-action problem for discrete distributions. Rates of convergence of the excess risk (the regret) of the EB rules are the main interest here. Previous results on the same problem have examined EB rules in the discrete exponential family or in particular types of discrete distributions. Here, we study EB rules under very general set-up, where the distributions of the observations are not fixed. When specialised to specific distributions, however, our results reduce to similar results available in the literature for such specific distributions. We show that the rates of convergence of the proposed EB rules are of the exponential order of the form O(exp(−cd n )), where {d n } is a sequence of positive numbers decreasing to zero as n→∞, with n being the number of observations. Another distinct feature of our results is that they are stated for a class of EB rules for the present problem, whereas some particular EB rules have been studied in previous work.