(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 n N

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