Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 − β among all possible tests of a given size α. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses.

Setting

Let $X$ denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions $f_{\theta}(x)$ , which depends on the unknown deterministic parameter $\theta \in \Theta$ . The parameter space $\Theta$ is partitioned into two disjoint sets $\Theta_0$ and $\Theta_1$ . Let $H_0$ denote the hypothesis that $\theta \in \Theta_0$ , and let $H_1$ denote the hypothesis that $\theta \in \Theta_1$ . The binary test of hypotheses is performed using a test function $\phi(x)$ .

meaning that $H_1$ is in force if the measurement $X \in R$ and that $H_0$ is in force if the measurement $X \in A$ . Note that $A \cup R$ is a disjoint covering of the measurement space.