Testing for Location-Parameter Family with Errors in Variables via Empirical Bayes Method

摘要

We study the empirical Bayes linear-loss two-action problem for the location-parameter family with errors in variables. Let p_0 = f_0 * q, the convolution of a frequency probability density f_0 and the probability density q of the error variable. Two empirical Bayes tests δ_n~* and δ_n are constructed depending on whether po has monotone likelihood ratio or not. It is shown that the rate of convergence of δ_n~* is either O(n/(ln n)~2) if po is supersmooth or O(n~(-(4α-3)/(4α-1)) if p_0 is ordinary smooth with parameter α ≥ 1; the rate of convergence of δ_n is shown to be either O(n/(ln n)~2~(δ/2)) if p_ 0 is supersmooth or O(n~(-(2α-1)/(4α)) if po is ordinary smooth, where 0 < δ < 1. It is concluded that both the MLR and supersmoothness have important impact on the performance of the empirical Bayes tests.