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  "body": "\n## Definition\n\nLet $Z_1, \\dots, Z_n$ be [[i.i.d.|0AJN7WHBJZ]] real random variables with a continuous distribution that is symmetric about a number $\\theta$, that is, $Z_1 - \\theta$ and $\\theta - Z_1$ have the same law.\nLet $R_i$ be the rank of $\\abs{Z_i}$ among $\\abs{Z_1}, \\dots, \\abs{Z_n}$, so that the smallest has rank $1$.\nThe *signed-rank statistic* is\n$$W^+ = \\sum_{i \\,:\\, Z_i \u003e 0} R_i.$$\nThe *Wilcoxon signed-rank test* of $H_0 \\colon \\theta = 0$ rejects $H_0$ when $W^+$ is too large, too small, or either, according to the alternative.\nUnder $H_0$, $W^+$ has the law of $\\sum_{k=1}^{n} k B_k$, where $B_1, \\dots, B_n$ are independent [[Bernoulli|0J97P5KZY0]] variables with parameter $1/2$.\n\n## Remarks\n\nUnder $H_0$, $E[W^+] = n(n+1)/4$ and $\\operatorname{Var}(W^+) = n(n+1)(2n+1)/24$, and $W^+$ is asymptotically normal.\nFor matched pairs $(X_i, Y_i)$ one takes $Z_i = Y_i - X_i$.\n\nExample: $Z = (1.2, -0.4, 2.5)$.\nThe absolute values $1.2, 0.4, 2.5$ have ranks $2, 1, 3$, so $W^+ = 2 + 3 = 5$.\nUnder $H_0$ each of the $8$ sign patterns has probability $1/8$, and only the sets of ranks $\\set{2, 3}$ and $\\set{1, 2, 3}$ give a sum at least $5$.\nSo the one-sided p-value is $P(W^+ \\ge 5) = 2/8 = 1/4$.\n\nNon-example: if the $Z_i$ have median $0$ but a skewed law, the clause on symmetry fails.\nThen the null law above is wrong, and the test can reject often even though the median is $0$; the [[sign test|NBK6G2VB7A]] is valid in that case.\n\nInverting the test gives the [[Hodges–Lehmann estimator|T6KRQAQQ15]] of $\\theta$.\n",
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