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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with [[empirical distribution function|A05WR6TEK5]] $F_n$.\nLet $F_0$ be a fixed continuous [[cumulative distribution function|0MXMS3AE13]].\nThe *Cramér–von Mises statistic* is\n$$T = n \\int_{-\\infty}^{\\infty} \\big(F_n(x) - F_0(x)\\big)^2 \\, dF_0(x).$$\nThe *Cramér–von Mises test* of $H_0 \\colon$ \"the $X_i$ have distribution function $F_0$\" rejects $H_0$ when $T$ is large.\nThe critical value comes from the null law of $T$.\n\n## Remarks\n\nWith the [[order statistics|58NHGZDSRE]] $X_{(1)} \\le \\dots \\le X_{(n)}$, the integral can be computed exactly:\n$$T = \\frac{1}{12 n} + \\sum_{i=1}^{n} \\Big( \\frac{2i - 1}{2n} - F_0(X_{(i)}) \\Big)^2.$$\nUnder $H_0$ the law of $T$ does not depend on $F_0$, since $F_0(X_i)$ is uniform on $[0, 1]$.\nAs $n \\to \\infty$ it converges to the law of $\\int_0^1 B(t)^2 \\, dt$ for a [[Brownian bridge|3MCVMBAGD1]] $B$.\n\nExample: $n = 1$, $X_1 = 0.5$ and $F_0(x) = x$ on $[0, 1]$.\nThe integral is $\\int_0^{1/2} x^2 \\, dx + \\int_{1/2}^{1} (1 - x)^2 \\, dx = 1/24 + 1/24 = 1/12$.\nThe formula gives the same value: $1/12 + (1/2 - 1/2)^2 = 1/12$.\n\nNon-example: $n \\int (F_n - F_0)^2 \\, dx$, with Lebesgue measure in place of $dF_0$, is not the criterion.\nIts null law depends on $F_0$, for example on its scale, so it gives no distribution-free test.\n\nThe Anderson–Darling statistic is a weighted variant of $T$.\n",
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