Let Z1,..., ZN be a sample from a distribution with density f (z ), where f

Let Z1,..., ZN be a sample from a distribution with density f (z −

θ), where f (z) is positive for all z and f is symmetric about 0, and let m, n, and the Sj be defined as in the preceding problem.

(i) The distribution of n and the Sj is given by P{the number of positive Z’s is n and S1 = s1,..., Sn = sn} (6.71)

= 1 2N E

 f V(r1) + θ

... f V(rm ) + θ

f V(s1) − θ

... f V(sn ) − θ

f V(1)

... f V(N)



, where V(1) < ··· < V(N), is an ordered sample from a distribution with density 2 f (v) for v > 0, and 0 otherwise.

(ii) The rank test of the hypothesis of symmetry with respect to the origin, which maximizes the derivative of the power function at θ = 0 and hence maximizes the power for sufficiently small θ > 0, rejects, under suitable regularity conditions, when

−E

⎡

⎣

n j=1 f

(V(sj)

f (V(sj)

⎤

⎦ > C.

(iii) In the particular case that f (z) is a normal density with zero mean, the rejection region of (ii) reduces to E(V(sj) > C, where V(1) < ··· < V(N) is an ordered sample from a χ-distribution 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 − θ) for sufficiently small positive θ.

[(i): Apply Problem 6.44(i) to find an expression for P{S1 = s1,..., Sn = sn given that the number of positive Z’s is n}.]