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

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} (6.68)

= 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 2f(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 6.42(i) to find an expression for P{S1 = s1,...,Sn = sn given that the number of positive Z’s is n}.]