Consider the rank sum test statistic from Example 11.2, which is a linear rank statistic with \(a(i)=i\) and \(c(i)=\delta\{i ;\{m+1, \ldots, n+m\}\}\) for all \(i=1, \ldots, n+m\). Under the null hypothesis that the shift parameter \(\theta\) is zero, find the mean and variance of this test statistic.

Example 11.2. Let X,..., X, be a set of independent and identically dis- tributed random variables from a continuous distribution F and let Y,..., Ym be a set of independent and identically distributed random variables from a continuous distribution G where G(t) = F(t) and is an unknown pa- rameter. This type of model for the distributions F and G is known as the shift model, and in the special case where the means of F and G both exist, 0=E(Y) E(X1). To test the null hypothesis Ho: 0 = 0 against the al- ternative hypothesis H:00 compute the ranks of the combined sample {X1,Xm, Y., Y}. Denote this combined sample as Z1,..., Zn+m and denote the corresponding ranks as R,..., Rn+m. Note that under the null hypothesis the combined sample can be treated as a single sample from the distribution F. Now consider the test statistic n+m Mm,n = Di Ri, i=1 where D = 0 when Z; is from the sample X,..., Xm and D; = 1 when Zi is from the sample Y,..., Y. Theorem 11.1 implies both that the random vector D = (D1,..., Dn+m)' is distribution free and that D is independent of the vector of ranks given by R. Hence, it follows that Mm,n = D'R is distribution free. A test based on Mm,n is known as the Wilcoxon, Mann, and Whitney Rank Sum test. The exact distribution of the test statistic under the null hypothesis can be derived by considering possible all equally likely configurations of D and R, and computing the value of Mm,n for each. See Chapter 4 of Hollander and Wolfe (1999) for further details.