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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 *Anderson–Darling statistic* is\n$$A^2 = n \\int_{-\\infty}^{\\infty} \\frac{\\big(F_n(x) - F_0(x)\\big)^2}{F_0(x)\\big(1 - F_0(x)\\big)} \\, dF_0(x).$$\nThe *Anderson–Darling test* of $H_0 \\colon$ \"the $X_i$ have distribution function $F_0$\" rejects $H_0$ when $A^2$ is large.\nThe critical value comes from the null law of $A^2$.\n\n## Remarks\n\nWith the [[order statistics|58NHGZDSRE]] $X_{(1)} \\le \\dots \\le X_{(n)}$ and $U_i = F_0(X_{(i)})$,\n$$A^2 = -n - \\frac{1}{n} \\sum_{i=1}^{n} (2i - 1) \\big( \\ln U_i + \\ln(1 - U_{n+1-i}) \\big).$$\nUnder $H_0$ the law of $A^2$ does not depend on $F_0$.\nWhen parameters of $F_0$ are estimated from the data, for example in a test of normality, other tables of critical values are used.\n\nExample: $n = 1$, $X_1 = 1/2$ and $F_0(x) = x$ on $[0, 1]$.\nThe integral is $\\int_0^{1/2} \\frac{x}{1 - x} \\, dx + \\int_{1/2}^{1} \\frac{1 - x}{x} \\, dx = 2\\big(\\ln 2 - \\tfrac12\\big) = 2\\ln 2 - 1$.\nThe formula gives $-1 - 2\\ln(1/2) = 2 \\ln 2 - 1 \\approx 0.386$, the same value.\n\nNon-example: the [[Cramér–von Mises statistic|G14XJXPKFS]] $n \\int (F_n - F_0)^2 \\, dF_0$ lacks the weight $1/(F_0(1 - F_0))$, so it is a different statistic that gives less weight to the tails.\n",
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