Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

604 Nonparametric and Robust Statistics

Note that this is the same ARE as derived in the last section between the signed-

rank Wilcoxon and thet-test. Iff(x) is a normal pdf, then the MWW has efficiency

95.5% relative to the pooledt-test. Thus the MWW tests lose little efficiency at

the normal. On the other hand, it is much more efficient than the pooledt-test at

the family of contaminated normals (with>0), as in Example 10.3.3.

10.4.2 Estimating Equations Based on the Mann–Whitney–Wilcoxon

As with the signed-rank Wilcoxon procedure in the last section, we invert the test
statistic to obtain an estimate of Δ. As discussed in the next section, this esti-
mate can be defined in terms of minimizing a norm. The estimator̂θW solves the
estimating equations

U(Δ) =EH 0 (U)=

n 1 n 2

2

. (10.4.27)

Recalling the description of the processU(Δ) described above, it is clear that the
Hodges–Lehmann estimator is given by

Δ̂U=medi,j{Yj−Xi}. (10.4.28)

The asymptotic distribution of the estimate follows in the same way as in the last
section based on the processU(Δ) and the asymptotic power lemma, Theorem
10.4.2. We avoid sketching the proof and simply state the result as a theorem:

Theorem 10.4.3.Assume that the random variablesX 1 ,X 2 ,...,Xn 1 are iid with
pdff(x)and that the random variablesY 1 ,Y 2 ,...,Yn 2 are iid with pdff(x−Δ).
Then

Δ̂Uhas an approximateN

(

Δ,τW^2

(

1

n 1 +

1

n 2

))

distribution, (10.4.29)

whereτW=(

√

12

∫∞

−∞f

(^2) (x)dx)− (^1).
As Exercise 10.4.6 shows, provided the Var(εi)=σ^2 <∞, the LS estimate
Y−Xof Δ has the following approximate distribution:
Y−Xhas an approximateN
(
Δ,σ^2
(
1
n 1 +
1
n 2
))
distribution. (10.4.30)
Note that the ratio of the asymptotic variances ofΔ̂Uis given by the ratio (10.4.26).
Hence the ARE of the tests agrees with the ARE of the corresponding estimates.

10.4.3 Confidence Interval for the Shift Parameter Δ

The confidence interval for Δ corresponding to the MWW estimate is derived the

same way as the Hodges–Lehmann estimate in the last section. For a given level

α, let the integercdenote the critical point of the MWW distribution such that

PΔ[U(Δ)≤c]=α/2. As in Section 10.2.3, we then have

1 −α = PΔ[c<U(Δ)<n 1 n 2 −c]

= PΔ[Dc+1≤Δ<Dn 1 n 2 −c], (10.4.31)