The Generalized Likelihood Ratio Test

The Neyman-Pearson lemma is airtight, but only for a narrow question: one fully specified $\theta_0$ against one fully specified $\theta_1$. Almost nothing a quant desk actually tests looks like that. "Does this GARCH model need the leverage term at all?", "is the recovery rate the same across these two sectors?", "does a three-regime model fit better than two?" — every one of these pits a composite null against a composite alternative, sets of parameter values rather than single points, exactly the case Remark 2 of the Neyman-Pearson lemma note flagged as outside the lemma's reach. This note builds the tool that fills that gap: the generalized likelihood ratio test (GLRT), which replaces "the ratio of two likelihoods at two fixed points" with "the ratio of the best-case likelihoods each hypothesis can offer" — a maximized likelihood under the null over a maximized likelihood under the full model. The construction gives up the Neyman-Pearson lemma's exact, provable optimality, but it is the single most widely applicable test-construction principle in statistics, playing the same organizing role for testing that the MLE plays for estimation.

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