Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

602 Nonparametric and Robust Statistics

horizontal axis and thenon the vertical axis is replaced byn 1 n 2 ;thatis,U(Δ) is a
decreasing step function of Δ that steps down one unit at each differenceDi,with
the maximum value ofn 1 n 2.
We can then proceed as in the last two sections to obtain properties of inference
based on the Wilcoxon. Let the integercαdenote the critical value of a levelα
test of the hypotheses (10.2.2) based on the statisticU; i.e.,α=PH 0 (U≥cα).
LetγU(Δ) =PΔ(U ≥cα), for Δ≥0, denote the power function of the test.
The translation property, Lemma 10.2.1, holds for the processU(Δ). Hence, as in
Theorem 10.2.1, the power function is a nondecreasing function of Δ. In particular,
the Wilcoxon test is an unbiased test for the one-sided hypotheses (10.4.4).

10.4.1 AsymptoticRelativeEfficiency

The asymptotic relative efficiency (ARE) of the Wilcoxon follows along similar lines
as for the sign test statistic in Section 10.2.1. Here, consider the sequence of local
alternatives given by

H 0 :Δ=0versusH 1 n:Δn=√δn, (10.4.14)

whereδ>0. We also assume that

n 1

n →λ^1 ,

n 2

n →λ^2 ,whereλ^1 +λ^2 =1. (10.4.15)

This assumption implies thatn 1 /n 2 →λ 1 /λ 2 ; i.e, the sample sizes maintain the

same ratio asymptotically.

To determine the efficacy of the MWW, consider the average

U(Δ) =

1

n 1 n 2

U(Δ). (10.4.16)

It follows immediately that

μU(0) =E 0 (U(0)) =^12 and σ^2 U(0) = 12 nn+1 1 n 2. (10.4.17)

Because the pairs (Xi,Yj) are iid we have

μU(Δn)=EΔn(U(0)) =E 0 (U(−Δn)) =P 0 (Y−X>−Δn). (10.4.18)

The independence ofXandYand the fact

∫∞

−∞F(x)f(x)dx=1/2gives

P 0 (Y−X>−Δn)=E 0 (P 0 [Y>X−Δn|X])

= E 0 (1−F(X−Δn))

=1−

∫∞

−∞

F(x−Δn)f(x)dx

=

1

2

+

∫∞

−∞

(F(x)−F(x−Δn))f(x)dx

≈

1

2

+Δn

∫∞

−∞

f^2 (x)dx, (10.4.19)