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  "body": "\n## Definition\n\nLet $X$ be data with [[likelihood function|05FSTXM92W]] $L(\\theta \\mid x)$ for $\\theta$ in a parameter set $\\Theta$, and let $\\Theta_0 \\subset \\Theta$.\nFor data $x$ with $\\sup_{\\theta \\in \\Theta} L(\\theta \\mid x) \u003e 0$, the *likelihood-ratio statistic* is\n$$\\lambda(x) = \\frac{\\sup_{\\theta \\in \\Theta_0} L(\\theta \\mid x)}{\\sup_{\\theta \\in \\Theta} L(\\theta \\mid x)}.$$\nThe *likelihood-ratio test* of $H_0 \\colon \\theta \\in \\Theta_0$ against $H_1 \\colon \\theta \\in \\Theta \\setminus \\Theta_0$ rejects $H_0$ when $\\lambda(x) \\le c$.\nThe constant $c \\in [0, 1]$ is chosen so that $\\sup_{\\theta \\in \\Theta_0} P_\\theta(\\lambda(X) \\le c) \\le \\alpha$, for a given level $\\alpha$.\n\n## Remarks\n\nSince $\\Theta_0 \\subset \\Theta$, one has $0 \\le \\lambda(x) \\le 1$.\nThe two suprema are the likelihoods at the restricted and the unrestricted [[maximum likelihood estimates|0FREPFKNS7]], when these exist.\nOne often uses $-2 \\log \\lambda(X)$, which is large when $\\lambda$ is small.\nExample: for $X_1, \\dots, X_n$ i.i.d. $\\mathcal{N}(\\mu, 1)$ and $\\Theta_0 = \\set{0}$, one finds $-2 \\log \\lambda = n \\bar X^2$.\nSo the test rejects when $\\abs{\\sqrt{n}\\, \\bar X}$ is large, which is the usual two-sided z-test.\nFor two simple hypotheses, the test is the most powerful test of the [[Neyman–Pearson lemma|0EE19CFEEF]].\n",
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