Neyman–Pearson lemma

In statistics, the Neyman–Pearson lemma describes the existence and uniqueness of the likelihood ratio as a uniformly most powerful test in certain contexts. It was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman–Pearson lemma is part of the Neyman–Pearson theory of statistical testing, which introduced concepts such as errors of the second kind, power function, and inductive behavior. The previous Fisherian theory of significance testing postulated only one hypothesis. By introducing a competing hypothesis, the Neyman–Pearsonian flavor of statistical testing allows investigating the two types of errors. The trivial cases where one always rejects or accepts the null hypothesis are of little interest but it does prove that one must not relinquish control over one type of error while calibrating the other. Neyman and Pearson accordingly proceeded to restrict their attention to the class of all ${\displaystyle \alpha }$ level tests while subsequently minimizing type II error, traditionally denoted by ${\displaystyle \beta }$. Their seminal paper of 1933, including the Neyman–Pearson lemma, comes at the end of this endeavor, not only showing the existence of tests with the most power that retain a prespecified level of type I error (${\displaystyle \alpha }$), but also providing a way to construct such tests. The Karlin-Rubin theorem extends the Neyman–Pearson lemma to settings involving composite hypotheses with monotone likelihood ratios.