In
statistics, the Neyman–Pearson lemma 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 like
errors of the second kind,
power function, and inductive behavior.
[The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?: Journal of the American Statistical Association: Vol 88, No 424]
The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?: Journal of the American Statistical Association: Vol 88, No 424
/ref>[Wald: Chapter II: The Neyman-Pearson Theory of Testing a Statistical Hypothesis]
Wald: Chapter II: The Neyman-Pearson Theory of Testing a Statistical Hypothesis
/ref>[The Empire of Chance]
The Empire of Chance
/ref> 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 level tests while subsequently minimizing type II error, traditionally denoted by . 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 (), 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.
Statement
Consider a test with hypotheses and , where the probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(or probability mass function) is for .
For any hypothesis test with rejection set , and any