Neyman–Pearson lemma

Builds on Statistical hypothesis test Likelihood function Type I and type II errors

Statement

Consider a hypothesis test of the simple null hypothesis $H_0 \colon \theta = \theta_0$ against the simple alternative $H_1 \colon \theta = \theta_1$ . Let $L(\theta \mid x)$ be the likelihood of the data $x$ . For $\eta > 0$ , let the test reject $H_0$ when $\Lambda(x) = \frac{L(\theta_0 \mid x)}{L(\theta_1 \mid x)} \le \eta,$ and let $\alpha = P_{\theta_0}(\Lambda(X) \le \eta)$ be its size. Then this likelihood-ratio test is a most powerful test of size $\alpha$ . Each other test with size at most $\alpha$ has at most the same power at $\theta_1$ .

Remarks

Consider all tests with a probability of a type I error at most $\alpha$ . Among them, the likelihood-ratio test has the smallest probability of a type II error. For a discrete law, an exact size $\alpha$ can need a randomized test on the boundary $\Lambda(x) = \eta$ . Example: for an i.i.d. $\mathcal{N}(\mu, 1)$ sample and $\mu_1 > \mu_0$ , the test rejects $H_0$ when $\bar{x}$ is large.

History

Jerzy Neyman and Egon Pearson introduced the lemma in a paper in 1933.

Reading path

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Independent sigma-algebras definition
Independence of random variables definition
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Statistic definition
Likelihood function definition
Statistical hypothesis test concept
Type I and type II errors definition
Neyman–Pearson lemma statement

Where this leads

1 page builds on this one.

Wald–Wolfowitz optimality of the sequential probability ratio test

Tag	`0EE19CFEEF`
URI	`m2://0EE19CFEEF`
Aliases	`neyman-pearson-lemma`
Kind	statement, lemma
Writing	W1 stated
MSC	62F03
Status	This copy of the site has no build data. Run `m2 status 0EE19CFEEF`.
Uses	Statistical hypothesis test, Likelihood function, Type I and type II errors
Same as	Wikidata Q1375989 Wikipedia Neyman–Pearson lemma

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Wald–Wolfowitz optimality of the sequential probability ratio test