Neyman–Pearson lemma

The Neyman–Pearson lemma states that, for testing two simple hypotheses, a likelihood-ratio test is most powerful among tests with the same significance level.

Likelihood-ratio test: A hypothesis test that rejects a null hypothesis when the likelihood ratio crosses a chosen threshold. The lemma identifies this test as most powerful at a fixed significance level.
Simple hypothesis: A statistical hypothesis that specifies a single probability distribution for the data. The lemma compares one fully specified null distribution with one fully specified alternative.
Jerzy Neyman: A Polish mathematician and statistician who helped establish modern frequentist statistical theory. Neyman co-developed the result and broader theory of hypothesis testing.
Binomial test: A hypothesis test for a success probability based on the number of successes in independent Bernoulli trials. For two fixed success probabilities, its likelihood ratio yields a most powerful rejection rule.
Likelihood principle: The principle that inferential conclusions should depend on observed data only through the likelihood function. The lemma uses likelihood ratios but proves a specific power-optimality claim, not this general principle.
Likelihood function: A function of model parameters that measures how well each parameter value explains observed data. The ratio of likelihoods under the two hypotheses determines the test statistic.
Statistical hypothesis test: A procedure that uses sample data to decide whether to reject a statistical hypothesis. The lemma is a precise result about optimal tests.
Egon Pearson: A British statistician known for foundational work on hypothesis testing and statistical decision procedures. Pearson co-developed the lemma with Neyman.
Poisson test: A hypothesis test concerning the rate parameter of a Poisson distribution. Two specified rates form simple hypotheses that can be compared by their likelihood ratio.
Bayesian hypothesis testing: A framework that compares hypotheses using prior probabilities and the posterior distribution. Bayesian tests incorporate prior odds rather than optimizing frequentist power at fixed size.

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