Rates of convergence of empirical Bayes tests for a normal mean

In the present paper, the EB two-action problem under the linear error loss is considered for the family of normal distributions with the location parameter. The purpose is to establish the upper and lower bounds for the risk. A monotone adaptive empirical Bayes test is constructed with the regret risk converging to zero at a rate of O(n^-1(ln n)^3/2). The lower bound for the risk of the form O(n^-1(ln n)^1/2(ln ln n)^-1) is derived. In the authorʹs opinion, the (ln n ln ln n) times difference between the lower and the upper bounds is due not to the fact that the estimator suggested in the paper is not optimal but to the fact that the lower bound is not exact.