(i) Let m and n be the numbers of negative and positive observations among Z1,..., ZN , and let S1 < ··· < Sn denote the ranks of the positive Z’s among |Z1|,... |ZN |. Consider the N + 1 2 N(N − 1) distinct sums Zi + Z j with i = j as well as i = j. The Wilcoxon signed-rank statistic Sj , is equal to the number of these sums that are positive.
(ii) If the common distribution of the Z’s is D, then E Sj 
= 1 2 N(N + 1) − N D(0) − 1 2 N(N − 1)

D(−z) d D(z).
[(i) Let K be the required number of positive sums. Since Zi + Z j is positive if and only if the Z corresponding to the larger of |Zi| and |Z j| is positive, K = N i=1 N j=1 Ui j where Ui j = 1 if Z j > 0 and |Zi| ≤ Z j and Ui j = 0 otherwise.]