Empirical Bayes linear loss hypothesis testing in a non-regular exponential family

Abstract

Abstract Asymptotically optimal empirical Bayes (EB) two-action linear loss hypothesis test procedures are developed for the nonregular translated exponential family with densities f(x¦θ) = k exp (− k(x − θ))I (x > θ) , k known positive constant. Rates of convergence for the procedures to asymptotic optimality are investigated. The prior distribution G of the parameter θ is completely unknown and unspecified with support in (a1, a2), − ∞ ⩽ a1 f(x) = ∫f(x¦θ) d G (θ) first to provide a sequence of uniformally strongly consistent estimators Gn of G and then use this to develop asymptotically optimal EB test procedures δn(X). X = Xn + 1 ∼ f(x), for the present (n + 1)st testing problem. It is shown that the difference between the actual (n + 1) stage Bayes risk of δn and the optimal risk (the minimum Bayes risk) R(G) is of the order o(1) as n → ∞ for every prior distribution G for which E|θ| O (n − r (r + 1) log log n) γ 2 for every G with E|θ| 2 (1 − γ) for some 0