One-sided empirical Bayes test for location parameter in Gamma distribution

In this paper, we devote to constructing the one-sided empirical Bayes (EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n−δs2s+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O({n^{ - \frac{{\delta s}}{{2s + 1}}}})$$\end{document}, where 1/2 ≤ δ < 1 and s > 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied.