Convergence Rates for Empirical Bayes Two-Action Problems I. Discrete Case

Abstract

Abstract : A sequence of decision problems is considered where for each problem the observation has discrete probability function of the form p(x) = h(x) beta (lambda) lambda to the power x, x = 0,1,2,..., and where lambda is selected independently for each problem according to an unknown prior distribution G(lambda). It is supposed that for each problem one of two possible actions (e.g., 'accept' or 'reject') must be selected. Under various assumptions about h(x) and G(lambda) the rate at which the risk of the nth problem approaches the smallest possible risk is determined for standard empirical Bayes procedures. It is shown that for most practical situations, the rate of convergence to 'optimality' will be at least as fast as L(n)/n where L(n) is a slowly varying function (e.g., log n). The rate cannot be faster than 1/n and this exact rate is achieved in some cases. Arbitrarily slow rates will occur in certain pathological situations. (Author)