On singh's conjecture on rate of convergence of empirical bayes tests

Abstract

This paper deals with an empirical Bayes test δ* n for testing H 0 : θ ≤ θ0 against H 1 : θ > θ0 in the positive exponential family f(x\\ θ) = c( θ) u(x) exp(−x/θ), x > 0, 0 < θ ≤ M < ∞, using a weighted quadratic error loss. We investigate the asymptotic optimality of the empirical Bayes test δ* n . It is shown that the regret Bayes risk of δ* n converges to zero at a rate of order O(n −1). This convergence rate provides a solution to the question, raised by Singh (1979) and Singh and Wei (1992), regarding the achievement of the best possible rate in empirical Bayes problems.