(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 2N(N −1) distinct sums



Zi+Zj 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 2N(N + 1) − ND(0) − 1 2N(N − 1) 

D(−z) dD(z).

[(i) Let K be the required number of positive sums. Since Zi + Zj is positive if and only if the Z corresponding to the larger of |Zi| and |Zj | is positive, K = N i=1

N j=1 Uij where Uij = 1 if Zj > 0 and |Zi| ≤ Zj and Uij = 0 otherwise.]