Simple and Composite Null UMP Tests

Recall that a test is deemed UMP at a level $\alpha$ if for any value $\theta$ of the parameter in the alternative hypothesis, the same test is UMP for the test of the simple hypothesis $H_0: \theta = \theta_0$ vs. $H_1: \theta = \theta_1$ . The Neyman-Pearson Lemma tells us that when testing $H_0: X \sim P_{\theta_1} \text{ vs. } H_1: X \sim P_{\theta_2}$ that the Likelihood Ratio Test is uniformly most powerful among all level $\alpha$ tests. We now seek to extend this test to the composite hypothesis setting. First note though that it is in general implausible to have a UMP test for a two-sided hypothesis. We will see more carefully later on why this is so, but it has to do with the fact that different tests are UMP depending on which side of the null parameter values the alternative lies. Thus the best we can do is to find sufficient conditions for UMP tests of 1-sided composite nulls and alternatives. The extension to this case is known as the Karlin-Rubin Theorem, and states the following:

$\text{Theorem.}$ If $p(x|\theta)$ has monotone likelihood ratio in a sufficient statistic $T(X)$ then the test $T(X) > c$ is UMP for testing the hypothesis $H_0: \theta = \theta_0$ vs. $H_1: \theta > \theta_0$ .