Optimal Rate of Convergence of Monotone Empirical Bayes Tests for a Normal Mean

摘要

This paper studies monotone empirical Bayes tests for a normal mean under a linear loss. The optimal rate of convergence of the monotone empirical Bayes tests is obtained. Applying a few techniques and using the non-uniform estimate of the remainder in the central limit theorem, we are able to construct a monotone empirical Bayes test and show that it achieves the best possible rate over a broad class of prior distributions, while the best possible rate is obtained through an idea of Donoho and Liu by constructing the 'hardest two- point subproblem'. This answers the question raised recently by Karunamuni and Liang. The result indicates that n(exp -1) may not be an attainable lower bound for the monotone empirical Bayes tests in the continuous one- parameter exponential family. A method to construct the monotone empirical Bayes test achieving the optimal rate is also discussed in this paper.