CONVERGENCE RATES FOR EMPIRICAL BAYES TWO-ACTION PROBLEMS I. DISCRETE CASE.

A sequence of decision problems is considered where for each problem the observation has discrete probability function of the form px hx 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 Glambda. It is supposed that for each problem one of two possible actions e.g., accept or reject must be selected. Under various assumptions about hx and Glambda 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 Lnn where Ln is a slowly varying function e.g., log n. The rate cannot be faster than 1n and this exact rate is achieved in some cases. Arbitrarily slow rates will occur in certain pathological situations. Author