14. The Wilcoxon rank-sum statistic can be represented as W  R1  R2  . . .  Rm, where Ri is the rank of Xi0 among all mn such differences. When H0 is true, each Ri is equally likely to be one of the rst mn positive integers; that is, Ri has a discrete uniform distribution on the values 1, 2, 3, . . . , m  n.

a. Determine the mean value of each Ri when H0 is true and then show that the mean value of W is m(m  n  1)/2. Hint: Use the hint given in Exercise 6(a).

b. The variance of each Ri is easily determined.

However, the Ri s are not independent random variables because, for example, if m  n  10 and we are told that R1  5, then R2 must be one of the other 19 integers between 1 and 20. However, if a and b are any two distinct positive integers between 1 and mn inclusive, P(Ria and Rj 

b)  1/[(m  n)(m  n  1)] since two integers are being sampled without replacement from among 1, 2, . . . ,mn. Use this fact to show that Cov(Ri, Rj)  (m  n  1)/12 and then show that the variance ofWis mn(mn1)/12.

c. A central limit theorem for a sum of nonindependent variables can be used to show that when m 8 and n 8, W has approximately a normal distribution with mean and variance given by the results of

(a) and (b). Use this to propose a large-sample standardized rank-sum test statistic and then describe the rejection region that has approximate signi cance level a for testing H0 against each of the three commonly encountered alternative hypotheses. Note:
When there are ties in the observed values, a correction for the variance derived in

(b) should be used in standardizing W; please consult a book on nonparametric statistics for the result.