10.5.∗General Rank Scores 617Other examples are given in the exercises.EXERCISES
10.5.1.In this section, as discussed above expression (10.5.2), the scoresaφ(i)are
generated by the standardized score functionφ(u); that is,
∫ 1∫^0 φ(u)du=0and
1
0 φ
(^2) (u)du= 1. Suppose thatψ(u) is a square-integrable function defined on the
interval (0,1). Consider the score function defined by
φ(u)=
ψ(u)−ψ
∫ 1
0 [ψ(v)−ψ]
(^2) dv
,
whereψ=
∫ 1
0 ψ(v)dv. Show thatφ(u) is a standardized score function.
10.5.2.Complete the derivation of the null variance of the test statisticWφby
showing the second term in expression (10.5.7) is true.
Hint:Use the fact that underH 0 ,forj =j′, the pair (aφ(R(Yj)),aφ(R(Yj′))) is
uniformly distributed on the pairs of integers (i, i′),i, i′=1, 2 ,...,n,i =i′.
10.5.3.For the Wilcoxon score functionφ(u)=
√
12[u−(1/2)], obtain the value of
sa. Then show that theVH 0 (Wφ) given in expression (10.5.8) is the same (except
for standardization) as the variance of the MWW statistic of Section 10.4.
10.5.4.Recall that the scores have been standardized so that
∫∞
−∞φ
(^2) (u)du=1.
Use this and a Riemann sum to show thatn−^1 s^2 a →1, wheres^2 a is defined in
expression (10.5.6).
10.5.5.Show that the normal scores, (10.5.29), derived in Example 10.5.1 are
standardized; that is,
∫ 1
0 φN(u)du=0and
∫ 1
0 φ
2
N(u)du=1.
10.5.6.∑ In Theorem 10.5.1, show that the minimum value ofWφ(Δ) is given by
n 2
j=1aφ(j) and that it is nonpositive.
10.5.7.Show thatEΔ[Wφ(0)] =E 0 [Wφ(−Δ)].
10.5.8. Consider the hypotheses (10.4.4). Suppose we select the score function
φ(u) and the corresponding test based onWφ. Suppose we want to determine the
sample sizen=n 1 +n 2 for this test of significance levelαto detect the alternative
Δ∗with approximate powerγ∗. Assuming that the sample sizesn 1 andn 2 are the
same, show that
n≈
(
(zα−zγ∗)2τφ
Δ∗
) 2
. (10.5.38)
10.5.9.In the context of this section, show the following invariances:
(a)Show that the parameterτφ, (10.5.24), is a scale functional as defined in
Exercise 10.1.4.