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 {displaystyle 1-eta } 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. In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power 1 − β {displaystyle 1-eta } 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 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_{ heta }(x)} , which depends on the unknown deterministic parameter θ ∈ Θ {displaystyle heta 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 heta in Theta _{0}} , and let H 1 {displaystyle H_{1}} denote the hypothesis that θ ∈ Θ 1 {displaystyle heta in Theta _{1}} .The binary test of hypotheses is performed using a test function φ ( x ) {displaystyle varphi (x)} . meaning that H 1 {displaystyle H_{1}} is in force if the measurement X ∈ R {displaystyle Xin R} and that H 0 {displaystyle H_{0}} is in force if the measurement X ∈ A {displaystyle Xin A} .Note that A ∪ R {displaystyle Acup R} is a disjoint covering of the measurement space. 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