An empirical Bayes test for testing $\vartheta \leq \vartheta_0$ against $\vartheta > \vartheta_0$ for the uniform distribution on $\lbrack 0, \vartheta)$ is discussed. The relation is shown with the estimation of a decreasing density on $\lbrack 0, \infty)$ and a monotone empirical Bayes test is derived based on the least-concave majorant of the empirical distribution function. The asymptotic distribution of the Bayes risk is obtained and some Monte Carlo results are given.
"Monotone Empirical Bayes Test for Uniform Distributions Using the Maximum Likelihood Estimator of a Decreasing Density." Ann. Statist. 15 (2) 875 - 879, June, 1987. https://doi.org/10.1214/aos/1176350381