In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 − β {\displaystyle 1-\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.
Setting Let X {\displaystyle X} denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions f θ ( x ) {\displaystyle f_{\theta }(x)} , which depends on the unknown deterministic parameter θ ∈ Θ {\displaystyle \theta \in \Theta } . The parameter space Θ {\displaystyle \Theta } is partitioned into two disjoint sets Θ 0 {\displaystyle \Theta _{0}} and Θ 1 {\displaystyle \Theta _{1}} . Let H 0 {\displaystyle H_{0}} denote the hypothesis that θ ∈ Θ 0 {\displaystyle \theta \in \Theta _{0}} , and let H 1 {\displaystyle H_{1}} denote the hypothesis that θ ∈ Θ 1 {\displaystyle \theta \in \Theta _{1}} . The binary test of hypotheses is performed using a test function φ ( x ) {\displaystyle \varphi (x)} with a reject region R {\displaystyle R} (a subset of measurement space).
φ ( x ) = { 1 if x ∈ R 0 if x ∈ R c {\displaystyle \varphi (x)={\begin{cases}1&{\text{if }}x\in R\\0&{\text{if }}x\in R^{c}\end{cases}}}
meaning that H 1 {\displaystyle H_{1}} is in force if the measurement X ∈ R {\displaystyle X\in R} and that H 0 {\displaystyle H_{0}} is in force if the measurement X ∈ R c {\displaystyle X\in R^{c}} . Note that R ∪ R c {\displaystyle R\cup R^{c}} is a disjoint covering of the measurement space.
Formal definition A test function φ ( x ) {\displaystyle \varphi (x)} is UMP of size α {\displaystyle \alpha } if for any other test function φ ′ ( x ) {\displaystyle \varphi '(x)} satisfying
sup θ ∈ Θ 0 E [ φ ′ ( X ) | θ ] = α ′ ≤ α = sup θ ∈ Θ 0 E [ φ ( X ) | θ ] {\displaystyle \sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi '(X)|\theta ]=\alpha '\leq \alpha =\sup _{\theta \in \Theta _{0}}\;\operatorname {E} [\varphi (X)|\theta ]\,}
we have
∀ θ ∈ Θ 1 , E [ φ ′ ( X ) | θ ] = 1 − β ′ ( θ ) ≤ 1 − β ( θ ) = E [ φ ( X ) | θ ] . {\displaystyle \forall \theta \in \Theta _{1},\quad \operatorname {E} [\varphi '(X)|\theta ]=1-\beta '(\theta )\leq 1-\beta (\theta )=\operatorname {E} [\varphi (X)|\theta ].}
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