Neyman-Pearson lemma

The Neyman-Pearson lemma is a fundamental result in statistical hypothesis testing that establishes the optimality of the likelihood ratio test. Formulated by Jerzy Neyman and Egon Pearson in 1933, the lemma proves that among all possible tests of a simple null hypothesis against a simple alternative, the test based on the likelihood ratio maximizes power for any given size (Type I error rate).

The lemma is not merely a technical theorem; it is the mathematical foundation of the Neyman-Pearson framework, which treats hypothesis testing as a decision problem with explicitly controlled error rates. It provided the rigorous justification for using likelihood ratios that Ronald Fisher had employed more intuitively, but with a crucial difference: Fisher used likelihood to measure evidence, while Neyman and Pearson used it to optimize decisions.

The lemma's limitation is that it applies only to simple hypotheses — fully specified distributions with no free parameters. The extension to composite hypotheses requires additional assumptions and spawned the field of uniformly most powerful tests.