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  "body": "\n## Statement\n\nLet $\\Theta \\subseteq \\R$, let $X$ have a density $f_\\theta$ with $\\theta \\in \\Theta$, and assume the family has a [[monotone likelihood ratio|4Z73AFN7WT]] in a [[statistic|036756NFD6]] $T(X)$.\nLet $\\theta_0 \\in \\Theta$ and $\\alpha \\in (0, 1)$.\nFor constants $c \\in \\R$ and $\\gamma \\in [0, 1]$, let $\\varphi(x) = 1$ if $T(x) \u003e c$, $\\varphi(x) = \\gamma$ if $T(x) = c$, and $\\varphi(x) = 0$ if $T(x) \u003c c$.\nThen $c$ and $\\gamma$ can be chosen so that $E_{\\theta_0}[\\varphi(X)] = \\alpha$.\nWith this choice, $\\varphi$ is a [[uniformly most powerful test|QRAPTPVH8C]] at level $\\alpha$ of $H_0 \\colon \\theta \\le \\theta_0$ against $H_1 \\colon \\theta \u003e \\theta_0$.\n\n## Remarks\n\nProof idea: fix $\\theta_1 \u003e \\theta_0$.\nBy the monotone likelihood ratio, $\\varphi$ rejects exactly when $f_{\\theta_1}/f_{\\theta_0}$ is large, so it is the most powerful test of $\\theta_0$ against $\\theta_1$ by the [[Neyman–Pearson lemma|0EE19CFEEF]].\nThis test does not depend on $\\theta_1$.\nOne then shows that its [[power function|4RJSN0QARD]] is nondecreasing in $\\theta$, so its type I error is at most $\\alpha$ on all of $\\set{\\theta \\le \\theta_0}$.\nExample: for $X$ binomial with parameters $n$ and $p$, the UMP test of $p \\le p_0$ against $p \u003e p_0$ rejects for a large number of successes, with randomization at the boundary value.\nThe theorem is about one-sided hypotheses; for $H_1 \\colon \\theta \\ne \\theta_0$ a UMP test usually does not exist.\n\n## Explanations\n\n### L1 human\n\nWhen larger values of a statistic always point towards larger parameters, the best one-sided test is the obvious one.\nIt rejects \"the parameter is at most $\\theta_0$\" when the statistic is large.\nThe theorem says that this simple test is the best possible against every larger parameter at once.\n\n### L2 human\n\nThe Karlin–Rubin theorem extends the Neyman–Pearson lemma from a simple alternative to a one-sided composite alternative.\nThe key is that the most powerful test against one alternative is the same for all alternatives on one side.\nIt applies to one-parameter exponential families, such as binomial, Poisson and normal models with one unknown parameter.\nFor two-sided problems, one restricts to unbiased tests to get an optimal test.\nFor discrete data, the randomization $\\gamma$ is needed to reach the exact level $\\alpha$.\n",
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  "name": "Karlin–Rubin theorem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/RCA/RCASN31FFS.md",
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