An empirical bayes two action problem with nonidentical components for a translated exponential distribution

Abstract

Let θ be a one dimensional random variable with distribution G with support in (0,∞) and let an observation be taken from a population with density is a known constant. A decision is to be made between where c is a known constant with loss function . Given that the preceding problem occurs repeatedly and independently with G unknown and K varying in from problem to problem, an empirical Bayes procedure is exhibited whose expected risk less a minimum Bayes risk is 0(n −θ;/4) whenever . The procedure depends on the estimation of a marginal density h and a weighted integral of h using the inversion formula for some absolutely integrable characteristic functions. Extension of these results to the case of a two-dimensional random vector (θ1, δ2) is also discussed.