On Empirical Bayes Testing for a Location Parameter in a Shifted Gamma Distribution

摘要

Consider a location parameter θ in a shifted gamma distribution having pdf f(x∣θ) = (x-θ)~(α-1) exp{-(x-θ)}/Γ(α)I_((θ, ∞)) (x), where α ≥ 2 is the fixed shape parameter. We study the two-action problem of testing H_0:θ≤θ_0 against H_1: θ>θ_0 using the empirical Bayes approach. An empirical Bayes test δ_n~* is constructed. Under some mild conditions, δ_n~* is shown to be asymptotically optimal at a rate of order O(n~(-2(α-1)/(2α-1))) for α ≥ 2, where n is the number of past data available when the current decision problem is considered. The achieved rate of δ_n~* much improves the existing result in the literature.