Empirical Bayes Testing for a Nonexponential Family Distribution

Abstract

This article considers the empirical Bayes testing problem for the parameter θ in a nonexponential family distribution having probability density f(x|θ) = k(θ)e −x /(1 + θ x), x > 0, θ > 0, and k(θ) > 0 using a linear error loss. An empirical Bayes test is proposed and its corresponding asymptotic optimality is investigated. Under the condition that E[θ r ] < ∞ for some positive integer r ≥ 2, and the critical point of a Bayes test is finite, is shown to be asymptotically optimal at a rate n −2r/(2r+1) where n is the number of past data available when the current testing problem is considered.