8.2 A single random sample D1, D2, . . . , DN of size N is drawn from a population which is continuous and symmetric. Assume there are m positive values, n negative values, and no zero values. Define the mþ n¼N random variables Xi ¼ Di if Di > 0 Yi ¼ jDij if Di < 0 Then the X1, X2, . . . , Xm and Y1, Y2, . . . , Yn constitute two independent random samples of sizes m and n.

(a) Show that the two-sample Wilcoxon rank-sum statistic WN of (8.2.1)
for these two samples equals the Wilcoxon signed-rank statistic Tþ
defined in (5.7.1).

(b) If these two samples are from identical populations, the median of the symmetric D population must be zero. Therefore the null distribution of WN is identical to the null distribution of Tþ conditional upon the observed number of plus and minus signs. Explain fully how tables of the null distribution of WN could be used to find the null distribution of Tþ. Since for N large, m and n will both converge to the constant value N=2 in the null case, these two test statistics have equivalent asymptotic properties.