Optimal rate of convergence of monotone empirical Bayes tests for normal means

This paper studies monotone empirical Bayes tests (MEBTs) for N(θ,1) under a linear loss. The purpose is to give a complete answer to an open problem raised by Karunamuni [1996. Optimal rates of convergence of empirical Bayes tests for the continuous one-parameter exponential family. Ann. Statist. 24, 212-231] and Liang [2000. On an empirical Bayes test for a normal mean. Ann. Statist. 28, 648-655] on the optimal rate of MEBTs. Through a novel construction of “hardest 2-point subproblems”, a lower bound rate O(n-1(lnn)1.5) is derived. This lower bound rate is shown to be achievable and therefore it is optimal.