Signed-Rank Wilcoxon Test

Definition

Let $X_1,X_2,\dots ,X_n$ be Random Sample from cdf $F(x)$, symmetric pdf $f(x)$, and median $\theta$. Consider a Sign Test $H_0:\theta =0,H_1:\theta >0$

The signed-rank Wilcoxon test statistic for the test is defined as $T=\sum _{i=1}^n\mathrm{sgn}(X_i)R|X_i|$ where $R|X_i|$ is the rank of $X_i$, among $|X_i|,\dots ,|X_n|$

We reject $H_0:\theta =0$ if $T\ge c$, where $c$ is determined by the level $\alpha$

$R|X_i|$ takes one of $1,2,\dots ,n$. So, we can rewrite the statistic $T$ as $T=\sum _{i=1}^{n}j\cdot \mathrm{sgn}(X_{ij})$ where $X_{ij}$ is an observation such that $R|X_i|=j$

Let $T^{+}$ be the sum of the ranks of positive $X_i$‘s, then $T=2T^{+}-\frac{n(n+1)}{2}$ So. $T^{+}=\frac{T}{2}+\frac{n(n+1)}{4}$ $E(T^{+})=\frac{n(n+1)}{4},\mathrm{Va}{\mathrm{r}}_{{\mathrm{H}}_0}(T^{+})=\frac{n(n+1)(2n+1)}{24}$

$T^{+}$ can be written as $T^{+}=\underset{i\le j}{\sum \sum }I\left(\frac{X_i+X_j}{2}>0\right)$ where $\frac{X_i+X_j}{2}$ are called the Walsh averages

In general, if the median of $X_i$ is $\theta$, then $T^{+}=\underset{i\le j}{\sum \sum }I\left(\frac{X_i+X_j}{2}>\theta \right)$

Consider a sequence of local alternatives $H_0:\theta =0,H_1:{\theta }_n=\delta /\sqrt{n},\delta >0$ The efficacy of the modified signed-rank Wilcoxon test statistic $T^{+}$ is $c_{T^{+}}={\tau }_{\omega }^{-1}=\sqrt{12}{\int }_{-\infty }^{\infty }{f}^2(x)dx$

Facts

Assume that $f(x)$ is symmetric, then under $H_0:\theta =0$, $|X_1|,|X_2|,\dots ,|X_n|$ are independent of $\mathrm{sgn}(X_1),\mathrm{sgn}(X_2),\dots ,\mathrm{sgn}(X_n)$

Let $X_1,X_2,\dots ,X_n$ be Random Sample from cdf $F(x)$, symmetric pdf $f(x)$, and median $\theta$, and $T=\sum _{i=1}^n\mathrm{sgn}(X_i)R|X_i|$, then under $H_0:\theta =0$

• $T$ is distribution free
• $E_{H_0}(T)=0$
• ${\mathrm{Var}}_{H_0}(T)=\frac{n(n+1)(2n+1)}{6}$
• $\frac{T}{\sqrt{{\mathrm{Var}}_{H_0}(T)}}\stackrel{D}{\to }N(0,1)$

Consider a sequence of hypothesis $H_0:\theta =0,H_1:{\theta }_n=\delta /\sqrt{n},\delta >0$, then ${\lim }_{n\to \infty }{\gamma }_{SR}({\theta }_n)=1-\Phi (z_a-\delta {\tau }_{\omega }^{-1})$ where $\Phi$ is a CDF of standard normal distribution, and ${\tau }_{\omega }=c_{T^{+}}^{-1}=\frac{1}{\sqrt{12}{\int }_{-\infty }^{\infty }{f}^2(x)dx}$

The estimator ${\hat{\theta }}_W$, which can be a solution of the equation $T^{+}({\hat{\theta }}_W)=\frac{n(n+1)}{4}$, is called Hodges-Lehmann estimator. Hodges-Lehmann estimator is an estimator of median of the Warsh average $\frac{X_i+X_j}{2}$.

Let $X_i=\theta +{ϵ}_i$ be random variables from location model, where ${ϵ}_i$‘s are i.i.d. with CDF $F(x)$, symmetric PDF $f(x)$, and median 0, then $\sqrt{n}({\hat{\theta }}_W-\theta )\stackrel{D}{\to }N(0,{\tau }_W^2)$ where ${\tau }_W=\frac{1}{\sqrt{12}{\int }_{-\infty }^{\infty }{f}^2(x)dx}$