Monotone Empirical Bayes Tests for the Continuous One-Parameter Exponential Family

Abstract

Let $\theta$ be the natural parameter of a continuous one-parameter exponential family. An empirical Bayes test is constructed for testing $\theta \leqq 0$ against $\theta > 0$ with a piecewise linear loss function. Since the problem is monotone, Bayes tests for a given prior distribution can be characterized by a single parameter, e.g., the size of the test under $\theta = 0$. Therefore the construction of an empirical Bayes test can be reduced to the construction of an estimator of this parameter. Such an estimator is constructed and the convergence rate of its mean squared error is investigated. The empirical Bayes test constructed in this way has not only nice asymptotic properties, but it can also be applied to small samples because of its (weak) admissibility.