On a monotone empirical Bayes test in a positive exponential family

Abstract

This paper studies a monotone empirical Bayes test δ ^* _n for testing H ₀:θ≤θ ₀ against H ₁:θ>θ ₀ for the positive exponential family f(x|θ)=c(θ)u(x)exp (−x/θ), x>0, using a weighted quadratic error loss. We investigate the convergence rate of the empirical Bayes test δ ^* _n . It is shown that the regret of δ ^* _n converges to zero at a rate O(n ⁻¹), where n is the number of past data available when the present testing problem is considered. Errors regarding the rate of convergence claimed in Gupta and Li (J. Stat. Plan. Inference, 129: 3–18, 2005) are addressed.