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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] with density $f_\\theta$, where $\\theta$ lies in an open set $\\Theta \\subseteq \\R^k$.\nAssume the standard regularity conditions for the asymptotic normality of the maximum likelihood estimator, with a positive definite Fisher information.\nLet $\\Theta_0 \\subseteq \\Theta$ be the set of $\\theta$ whose first $q$ coordinates equal fixed values, with $1 \\le q \\le k$.\nLet $\\lambda_n$ be the [[likelihood-ratio statistic|YM1BNA80RC]] of $H_0 \\colon \\theta \\in \\Theta_0$ based on $X_1, \\dots, X_n$.\nIf the true $\\theta$ lies in $\\Theta_0$, then $-2 \\log \\lambda_n$ [[converges in distribution|0JDAQHWA9N]] to the [[chi-squared distribution|0RFMMBHFTV]] with $q$ degrees of freedom as $n \\to \\infty$.\n\n## Remarks\n\nThe number $q$ is the difference between the dimension of $\\Theta$ and the dimension of $\\Theta_0$.\nProof idea: expand the log-likelihood to second order around the unrestricted and the restricted maximum likelihood estimates.\nThe difference becomes a quadratic form in an asymptotically normal vector, which has a limiting chi-squared law with $q$ degrees of freedom.\nExample: for an i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ sample and $H_0 \\colon \\mu = 0$, here $k = 2$ and $q = 1$, so one rejects when $-2 \\log \\lambda_n$ exceeds the $1 - \\alpha$ quantile of $\\chi^2_1$.\nThe hypothesis that the true parameter is an interior point matters.\nFor a variance component tested at the boundary value $0$, the limit law is a mixture of $\\chi^2_0$ and $\\chi^2_1$ rather than $\\chi^2_1$.\n\n## Explanations\n\n### L1 human\n\nThe likelihood-ratio test needs a threshold, and for that one needs the law of its statistic when the null hypothesis holds.\nWilks' theorem says that for large samples this law is almost always the same: a chi-squared law.\nIts only parameter is the number of constraints that the null hypothesis imposes.\n\n### L2 human\n\nThis is why likelihood-ratio tests are used everywhere: one computes $-2 \\log \\lambda_n$ and compares it with a chi-squared quantile.\nThe score test and the Wald test have the same limiting chi-squared law under the null hypothesis, and the three tests are asymptotically equivalent.\nThe theorem needs regularity: smooth densities, identifiability, and a true parameter in the interior of the parameter set.\nIt is an asymptotic statement, so for small samples the chi-squared approximation can be poor.\n",
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