The Neyman-Pearson Lemma

The previous note built a working test for $H_0:\mu=0$ against $H_1:\mu>0$ by rejecting for large $\bar X$ — a sensible choice, but an unjustified one: nothing ruled out some other statistic doing better. The Neyman-Pearson lemma closes that gap completely, for the one setting where "best possible" can be pinned down exactly: a simple null against a simple alternative. It states, with no further assumptions on the model, that the test built from the likelihood ratio — reject $H_0$ whenever $f(x;\theta_1)/f(x;\theta_0)$ is large — is the most powerful test at its significance level, full stop, among every conceivable rejection rule with that level. This is one of the sharpest optimality results in all of statistics: it does not merely say the likelihood-ratio test is a good test, it says no test can ever beat it. This note states the result precisely and works it through the running normal-mean example; the next note proves it.

Learning objectives. By the end of this note you should be able to: