Wilcoxon signed-rank test

Builds on Statistical hypothesis test Independent and identically distributed random variables Bernoulli distribution

Definition

Let $Z_1, \dots, Z_n$ be i.i.d. 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. Let $R_i$ be the rank of $\abs{Z_i}$ among $\abs{Z_1}, \dots, \abs{Z_n}$ , so that the smallest has rank $1$ . The signed-rank statistic is $W^+ = \sum_{i \,:\, Z_i > 0} R_i.$ The 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. Under $H_0$ , $W^+$ has the law of $\sum_{k=1}^{n} k B_k$ , where $B_1, \dots, B_n$ are independent Bernoulli variables with parameter $1/2$ .

Remarks

Under $H_0$ , $E[W^+] = n(n+1)/4$ and $\operatorname{Var}(W^+) = n(n+1)(2n+1)/24$ , and $W^+$ is asymptotically normal. For matched pairs $(X_i, Y_i)$ one takes $Z_i = Y_i - X_i$ .

Example: $Z = (1.2, -0.4, 2.5)$ . The absolute values $1.2, 0.4, 2.5$ have ranks $2, 1, 3$ , so $W^+ = 2 + 3 = 5$ . Under $H_0$ 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 $5$ . So the one-sided p-value is $P(W^+ \ge 5) = 2/8 = 1/4$ .

Non-example: if the $Z_i$ have median $0$ but a skewed law, the clause on symmetry fails. Then the null law above is wrong, and the test can reject often even though the median is $0$ ; the sign test is valid in that case.

Inverting the test gives the Hodges–Lehmann estimator of $\theta$ .

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Wilcoxon signed-rank test definition

Tag	`9RPYB7S4HM`
URI	`m2://9RPYB7S4HM`
Kind	definition
Writing	W0 stub
MSC	62G10
Defines	Wilcoxon signed-rank test
Uses	Statistical hypothesis test, Independent and identically distributed random variables, Bernoulli distribution

Wilcoxon signed-rank test

Definition

Remarks

Reading path

Further reading

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