Mann-Whitney-Wilcoxon Procedure

Definition

Let $X_1,X_2,\dots ,X_{n_1}$ be Random Sample from a distribution with cdf $F(x)$, and $Y_1,Y_2,\dots ,Y_{n_2}$ be Random Sample from a distribution with cdf $F(x-\Delta )$ Consider a test $H_0:\Delta =0,H_1:\Delta >0$

The Mann-Whitney-Wilcoxon test statistic for the test is defined as $W=\sum _{j=1}^{n_2}R(Y_j)$ where $R(Y_j)$ is the rank of $Y_j$ in the combined (pooled) sample of size $n=n_1+n_2$

We reject $H_0:\Delta =0$ when $W\ge c$, where $c$ is defined by the level $\alpha$

$R(Y_j)=\sum _{i=1}^{n_1}I(X_i<Y_i)+\sum _{k=1}^{n_2}I(Y_k\le Y_j)$ Therefore, the Mann-Whitney-Wilcoxon test statistic is decomposed as $W=\sum _{j=1}^{n_2}R(Y_j)=\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}I(X_i<Y_j)+\sum _{k=1}^{n_2}\sum _{j=1}^{n_2}I(Y_k\le Y_j)=U+\frac{n_2(n_2+1)}{2}$ where $U:=\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}I(X_i<Y_j)$

So, We reject $H_0:\Delta =0$ when $U\ge c$ $E(U)=\frac{n_1{n}_2}{2}$

Also, the power function of the test is defined as $U(\Delta )=\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}I(X_i<Y_j-\Delta )=\sum _{i=1}^{n_1}\sum _{j=1}^{n_2}I(Y_j-X_i>\Delta )$

Consider a sequence of local alternatives $H_0:\Delta =0,H_1:{\Delta }_n=\delta /\sqrt{n},\delta >0$, and assume that $\frac{n_1}{n}\to {\lambda }_1,\frac{n_2}{n}\to {\lambda }_2$, where ${\lambda }_1+{\lambda }_2=1$ The efficacy of the modified Mann-Whitney-Wilcoxon test statistic $U$ is $c_U={\tau }_W^{-1}=\sqrt{12}\sqrt{{\lambda }_1{\lambda }_2}{\int }_{-\infty }^{\infty }{f}^2(x)dx$

Facts

Let $X_1,X_2,\dots ,X_{n_1}$ be Random Sample from a distribution with cdf $F(x)$, $Y_1,Y_2,\dots ,Y_{n_2}$ be Random Sample from a distribution with cdf $F(x-\Delta )$, and $W=\sum _{j=1}^{n_2}R(Y_j)$ then under $H_0:\Delta =0$

• $W$ is distribution free
• $E_{H_0}(W)=\frac{n_2(n+1)}{2}$
• ${\mathrm{Var}}_{H_0}(W)=\frac{n_1{n}_2(n+1)}{12}$
• $\frac{W-n_2(n+1)/2}{\sqrt{{\mathrm{Var}}_{H_0}(W)}}\stackrel{D}{\to }N(0,1)$

Consider a sequence of hypothesis $H_0:\Delta =0,H_1:{\Delta }_n=\delta /\sqrt{n},\delta >0$, then ${\lim }_{n\to \infty }{\gamma }_U({\Delta }_n)=1-\Phi (z_a-\sqrt{{\lambda }_1{\lambda }_2}\delta {\tau }_W^{-1})$ where $\Phi$ is a CDF of standard normal distribution, and ${\tau }_W=c_U^{-1}=\frac{1}{\sqrt{12}\sqrt{{\lambda }_1{\lambda }_2}{\int }_{-\infty }^{\infty }{f}^2(x)dx}$