(i) Let X1,...,Xm; Y1,...,Yn be i.i.d. according to a continuous distribution F, let the ranks of the Y ’s be S1 < ··· < Sn, and let T = h(S1) + ··· + h(Sn). Then if either m = n or h(s) + h(N + 1 − s) is independent of s, the distribution of T is symmetric about nN i=1 h(i)/N.

(ii) Show that the two-sample Wilcoxon and normal-scores statistics are symmetrically distributed under H, and determine their centers of symmetry.

[(i): Let S

i = N + 1 − Si, and use the fact that T
= h(S

j ) has the same distribution under H as T.]

Section 6.10