The previous note stated the Neyman-Pearson lemma and used it, but a claim this strong — that a specific test beats every other test at the same significance level — earns its keep only once the proof is actually seen. This derivation carries out that proof in full, using the test function formalism (a decision rule written as a -valued, or more generally -valued, function of the data) and a single clean inequality that compares the Neyman-Pearson test against an arbitrary competitor pointwise, before integrating. The whole argument compresses into one line once the right inequality is written down; the work is in seeing why that inequality is true and what it delivers once summed or integrated over the sample space.
Learning objectives. By the end of this note you should be able to:
- define a test function and express significance level and power as expectations of ;
- state and verify the key pointwise inequality the proof of the Neyman-Pearson lemma is built from;
- carry out the integration step that converts the pointwise inequality into a comparison of two tests' power;
- reproduce the sufficiency half of the proof (Theorem 1(a)) in full, and sketch the necessity half (Theorem 1(b));
- identify exactly where the proof uses and where it would break down without a well-defined significance level.
In this lesson
- Test functions
- The key pointwise inequality
- Integrating the inequality
- The necessity half
Before you start
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