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  "body": "\n## Definition\n\nLet $X$ be data whose law $P_\\theta$ depends on $\\theta \\in \\Theta$, let $\\Theta_0$ and $\\Theta_1$ be disjoint subsets of $\\Theta$, and let $\\alpha \\in (0, 1)$.\nA (possibly randomized) test $\\varphi$ of $H_0 \\colon \\theta \\in \\Theta_0$ against $H_1 \\colon \\theta \\in \\Theta_1$ has level $\\alpha$ if $E_\\theta[\\varphi(X)] \\le \\alpha$ for every $\\theta \\in \\Theta_0$.\nA test $\\varphi^*$ is *uniformly most powerful* (UMP) at level $\\alpha$ if:\n\n1. $\\varphi^*$ has level $\\alpha$;\n2. for every test $\\varphi$ of level $\\alpha$ and every $\\theta \\in \\Theta_1$, $E_\\theta[\\varphi^*(X)] \\ge E_\\theta[\\varphi(X)]$.\n\n## Remarks\n\nHere $E_\\theta[\\varphi(X)]$ is the [[power function|4RJSN0QARD]] of $\\varphi$.\nExample: let $X_1, \\dots, X_n$ be i.i.d. $\\mathcal{N}(\\theta, 1)$, $H_0 \\colon \\theta \\le 0$, $H_1 \\colon \\theta \u003e 0$.\nThe test that rejects when $\\sqrt{n}\\, \\bar X \u003e z_{1 - \\alpha}$, the $1 - \\alpha$ quantile of $\\mathcal{N}(0, 1)$, is UMP at level $\\alpha$.\nFor each $\\theta_1 \u003e 0$ it is the most powerful test of $\\theta = 0$ against $\\theta = \\theta_1$, by the [[Neyman–Pearson lemma|0EE19CFEEF]], and its power function is increasing.\nNon-example: for $H_0 \\colon \\theta = 0$ against $H_1 \\colon \\theta \\ne 0$ in the same model, no test satisfies clause 2.\nThe one-sided test above is most powerful against each $\\theta_1 \u003e 0$, but at each $\\theta_1 \u003c 0$ the mirror one-sided test has strictly more power.\n",
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