77. (i) Let XI' . . . , Xm ; YI , . . . , Y,. be LLd. according to a continuous distribution F, let the ranks of the Y's be SI < . .. < Sn' and let T = h(SI) + .. . +h(SII)' Then if either m = n or h(s) + h(N + 1 - s) is independent of s, the distribution of T is symmetric about nE~_lh(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: = N + 1 - S;, and use the fact that T' = 'Lh(Sj) has the same distribution under HasT.] Note. The following problems explore the relationship between pivotal quantities and equivariant confidence sets. For more details see Arnold (1984). Let X be distributed according Pe." , and consider confidence sets for 8 that are equivariant under a group G*, as in Section 11. If w is the set of possible 8-values, define a group Gon flEx w by g(8, x) = (gx, g8).