Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

594 Nonparametric and Robust Statistics

Table 10.3.2:AREs among the sign, the Signed-Rank Wilcoxon, and thet-Tests

for Contaminated Normals withσc= 3 and Proportion of Contamination

0.00 0.01 0.02 0.03 0.05 0.10 0.15 0.25

ARE(W, t) 0.955 1.009 1.060 1.108 1.196 1.373 1.497 1.616

ARE(S, t) 0.637 0.678 0.719 0.758 0.833 0.998 1.134 1.326

ARE(W, S) 1.500 1.487 1.474 1.461 1.436 1.376 1.319 1.218

is discussed in Exercises 10.3.7 and 10.3.8. Recall that we also show that the location

estimator based on the sign test could be obtained by inverting the test. Considering

this for the Wilcoxon, the estimatorθ̂W solves

T+(̂θW)=

n(n+1)

4

. (10.3.31)

Using the description of the function T+(θ) after its definition, (10.3.15), it
is easily seen that ̂θW =median{(Xi+Xj)/ 2 }; i.e., the median of the Walsh
averages. This is often called the Hodges–Lehmann estimator because of several
seminal articles by Hodges and Lehmann on the properties of this estimator; see
Hodges and Lehmann (1963).
The R functionwilcox.testcomputes the Hodges–Lehmann estimate. To il-
lustrate its computation, consider the Darwin data in Example 10.3.1. Let the R
vectordscontain the paired differences, Cross−Self. The R code segment given
bywilcox.test(ds,conf.int=T)then computes the Hodges–Lehmann estimate
to be 3.1375. So we estimate the difference in heights to be 3.1375 inches.
Once again, we can use practically the same argument that we used for the sign
process to obtain the asymptotic distribution of the Hodges–Lehmann estimator.
We summarize the result in the next theorem.

Theorem 10.3.3. Consider a random sampleX 1 ,X 2 ,X 3 ,...,Xnwhich follows
Model (10.2.1). Suppose thatf(x)is symmetric about 0. Then

√

n(θ̂W−θ)→N(0,τW^2 ), (10.3.32)

whereτW=

(√

12

∫∞

−∞f

(^2) (x)dx
)− 1
Using this theorem, the AREs based on asymptotic variances for the signed-rank
Wilcoxon are the same as those defined above.

10.3.3 Confidence Interval for the Median

Because of the similarity between the processesS(θ)andT+(θ), confidence intervals
forθbased on the signed-rank Wilcoxon follow the same way as do those based on
S(θ). For a given levelα,letcW 1 , an integer, denote the critical point of the signed-
rank Wilcoxon distribution such thatPθ[T+(θ)≤cW 1 ]=α/2. As in Section 10.2.3,