uniformly most powerful test

In hypothesis testing, for a given $\alpha$, a uniformly most powerful (UMP) test is a test whose type-I error is bounded by $\alpha$ and, among all other tests with type-I error bounded by $\alpha$, has the most power.

More formally, suppose we are testing $H_0$ vs $H_1$. Consider the family of all tests $\psi$ such that ${\sup }_{P\in H_0}P(\psi (X)=1)\le \alpha$. We say a test $\varphi$ is UMP if for all other tests $\psi$, we have

$$P(\varphi (X)=1)\ge P(\psi (X)=1)\text{for all}P\in H_1.$$

The Neyman-Pearson lemma states that the likelihood-ratio test is UMP for testing simple hypothesis (i.e. simple null vs simple alternative). The Karlin-Rubin theorem extends this to composite vs composite but under a monotone likelihood ratio assumption.

UMP tests don’t always exist.