Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power ${\displaystyle \beta }$ 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.

Let ${\displaystyle X}$ denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions ${\displaystyle f_{\theta }(x)}$, which depends on the unknown deterministic parameter ${\displaystyle \theta \in \Theta }$. The parameter space ${\displaystyle \Theta }$ is partitioned into two disjoint sets ${\displaystyle \Theta _{0}}$ and ${\displaystyle \Theta _{1}}$. Let ${\displaystyle H_{0}}$ denote the hypothesis that ${\displaystyle \theta \in \Theta _{0}}$, and let ${\displaystyle H_{1}}$ denote the hypothesis that ${\displaystyle \theta \in \Theta _{1}}$. The binary test of hypotheses is performed using a test function ${\displaystyle \phi (x)}$.

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