Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.5.∗General Rank Scores 611

Suppose thatF̂n 1 andF̂n 2 are the empirical cdfs of the random samplesX 1 ,...,Xn 1
andY 1 ,...,Yn 2 , respectively. The relationship between the ranks and empirical cdfs
follows as

R(Yj+Δ) = #k{Yk+Δ≤Yj+Δ}+#i{Xi≤Yj+Δ}

=#k{Yk≤Yj}+#i{Xi≤Yj+Δ}

= n 2 F̂n 2 (Yj)+n 1 F̂n 1 (Yj+Δ). (10.5.14)

Substituting this last expression into expression (10.5.13), we get

μφ(Δ) =

1

n

∑n^2

j=1

E 0

{

φ

[

n 2

n+1

F̂n 2 (Yj)+ n^1

n+1

F̂n 1 (Yj+Δ)

]}

(10.5.15)

→ λ 2 E 0 {φ[λ 2 F(Y)+λ 1 F(Y+Δ)]} (10.5.16)

= λ 2

∫∞

−∞

φ[λ 2 F(Y)+λ 1 F(Y+Δ)]f(y)dy. (10.5.17)

The limit in expression (10.5.16) is actually a double limit, which follows from
F̂ni(x)→F(x),i=1,2, underH 0 , and the observation that upon substitutingFfor
the empirical cdfs in expression (10.5.15), the sum contains identically distributed
random variables and, thus, the same expectation. These approximations can be
made rigorous. It follows immediately that

d

dΔ

μφ(Δ) =λ 2

∫∞

−∞

φ′[λ 2 F(Y)+λ 1 F(Y+Δ)]λ 1 f(y+Δ)f(y)dy.

Hence

μ′φ(0) =λ 1 λ 2

∫∞

−∞

φ′[F(y)]f^2 (y)dy. (10.5.18)

From (10.5.12),

√

nσφ=

√

n

√

n 1 n 2

n(n−1)

1

√

n

√

1

n

s^2 a→

√

λ 1 λ 2. (10.5.19)

Based on (10.5.18) and (10.5.19), the efficacy ofWφis given by

cφ= lim

n→∞

μ′φ(0)

√

nσφ

=

√

λ 1 λ 2

∫∞

−∞

φ′[F(y)]f^2 (y)dy. (10.5.20)

Using the efficacy, the asymptotic power can be derived for the test statistic
Wφ. Consider the sequence of local alternatives given by (10.4.14) and the levelα
asymptotic test based onWφ. Denote the power function of the test byγφ(Δn).
Then it can be shown that

lim

n→∞

γφ(Δn)=1−Φ(zα−cφδ), (10.5.21)

where Φ(z) is the cdf of a standard normal random variable. Sample size deter-
mination based on the test statisticWφproceeds as in the last few sections; see
Exercise 10.5.8.