Convergence Rates for Empirical Bayes Two-Action Problems II. Continuous Case
Abstract : A sequence of decision problems is considered where for each problem the observation has a probability density function of exponential type with parameter lambda where lambda is selected independently for each problem according to an unknown prior distribution G(lambda). It is supposed that in each of the problems, one of two possible actions (e.g., 'accept' or 'reject') must be taken. Under various assumptions, reasonably sharp upper bounds are found for the rate at which the risk of the nth problem approaches the smallest possible risk for certain refinements of the standard empirical Bayes procedures. For suitably chosen procedures, under situations likely to occur in practice, rates faster than n to the power (-1 + epsilon) may be obtained for arbitrarily small epsilon > 0. Arbitrarily slow rates can occur in pathological situations. (Author)