Karlin–Rubin theorem

Remarks

Proof idea: fix $\theta_1 > \theta_0$. By the monotone likelihood ratio, $\varphi$ rejects exactly when $f_{\theta_1}/f_{\theta_0}$ is large, so it is the most powerful test of $\theta_0$ against $\theta_1$ by the Neyman–Pearson lemma. This test does not depend on $\theta_1$. One then shows that its power function is nondecreasing in $\theta$, so its type I error is at most $\alpha$ on all of $\set{\theta \le \theta_0}$. Example: for $X$ binomial with parameters $n$ and $p$, the UMP test of $p \le p_0$ against $p > p_0$ rejects for a large number of successes, with randomization at the boundary value. The theorem is about one-sided hypotheses; for $H_1 \colon \theta \ne \theta_0$ a UMP test usually does not exist.

Going deeper

For a student

The Karlin–Rubin theorem extends the Neyman–Pearson lemma from a simple alternative to a one-sided composite alternative. The key is that the most powerful test against one alternative is the same for all alternatives on one side. It applies to one-parameter exponential families, such as binomial, Poisson and normal models with one unknown parameter. For two-sided problems, one restricts to unbiased tests to get an optimal test. For discrete data, the randomization $\gamma$ is needed to reach the exact level $\alpha$.