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Probability, Statistics and Financial Data / Likelihood Ratio Tests, Wilks' Theorem, and Goodness of Fit

Overview — Likelihood Ratio Tests, Wilks' Theorem, and Goodness of Fit

Neyman-Pearson theory proved something remarkable and then immediately ran into its own limits: the Neyman-Pearson lemma delivers a provably unbeatable test, but only for a simple null against a simple alternative — one fixed parameter value against another. Almost nothing a quant desk tests actually looks like that. "Does this five-factor model need its fifth factor?", "did the default-intensity regime change last quarter?", "does the P&L distribution this model assumes actually match five years of realised outcomes?" — every one of these is a composite question, comparing whole sets of parameter values, or asking whether a parametric family fits at all. This topic builds the machinery for exactly that class of question, trading the Neyman-Pearson lemma's sharp, finite-sample optimality for a different kind of guarantee: asymptotic, but essentially universal across every regular parametric model a desk is likely to fit.
The generalized likelihood ratio test is the natural extension of the likelihood-ratio idea to composite hypotheses. Wilks' theorem is what makes it usable in practice: it says the test statistic's null distribution converges to a χ2\chi^2 distribution whose degrees of freedom are a pure counting exercise — free parameters lost to the restriction — regardless of which specific model is being tested. The Wald and score tests are two computationally cheaper large-sample alternatives, each needing only one of the two model fits the likelihood-ratio test requires, and each asymptotically equivalent to it. The topic closes with goodness-of-fit testing, taking this whole machinery outside the "test one parameter" setting entirely and asking the more basic question of whether an assumed distributional family is the right one to be testing inside at all — the Pearson chi-squared and Kolmogorov-Smirnov tests that answer it for categorical and continuous data respectively.

By the end of this topic you should be able to:

  • construct the generalized likelihood ratio statistic Λ(x)\Lambda(x) for a composite null and reduce it to a sufficient statistic;
  • state Wilks' theorem, compute its degrees of freedom by counting free parameters, and explain the second-order Taylor-expansion argument that proves it;
  • construct the Wald and score test statistics as computationally cheaper large-sample equivalents of the likelihood ratio test, and choose between all three based on computational cost, reparametrisation invariance, and boundary robustness;
  • apply Pearson's chi-squared test to binned categorical data, correctly accounting for estimated parameters in the degrees of freedom;
  • apply the Kolmogorov-Smirnov test to continuous data, and correct it with the Lilliefors adjustment when the reference distribution's parameters are estimated from the same sample.

In this lesson

  1. Contents
  2. The big picture

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