43. Let Z Z be a sample from a distribution with density f(z - ), where (z) is positive for all z and f is symmetric about 0, and let m, n, and the S, be defined as in the preceding problem. (i) The distribution of n and the S, is given by (66) P(the number of positive Z's is n and S...... = ... S = s,} 1 [ ( V (r) + 0)... ( V (rm) + 0) (V () - 0)... (V(s)- - 2N f(V)...(VN))

where V(l) < . .. < V( N) is an ordered sample from a distribution with density 2f( v) for v > 0, and ° otherwise. (ii) The rank test of the hypothesis of symmetry with respect to the origin, which maximizes the derivative of the power function at 8 = ° and hence maximizes the power for sufficiently small 8 > 0, rejects, under suitable regularity conditions, when [ n 1'( V(s) ] -E L > C. } -I f( V( Sj)) (iii) In the particular case that f(z) is a normal density with zero mean, the rejection region of (ii) reduces to I:E(V(s) > C, where V(l) < .. . < V( N) is an ordered sample from a x-di~tribution with 1 degree of freedom. (iv) Determine a density f such that the one-sample Wilcoxon test is most powerful against the alternatives f(z - 8) for sufficiently small positive 8. [(i): Apply Problem 29(i) to find an expression for P{ SI = Sl" ' " S; = s" given that the number of positive Z's is n }.]