596 Nonparametric and Robust StatisticsAs sketched in Exercise 10.3.2, under the assumptions of symmetry and location
estimators, MSEjis a consistent estimator of the variance of̂θjfor a sample of sizen. Hence, the estimate of the relative efficiency (REn) between the estimatorsθ̂ 1
and̂θ 2 at sample sizenis the ratio
RÊ
n(̂θ 1 ,θ̂ 2 )=MSE 2
MSE 1. (10.3.37)
To illustrate this discussion, consider a study comparing the Hodges–Lehmann
and sample mean estimators over the family of contaminated normal distributions
with rate of contamination and the standard deviation ratioσc, where we are using
the notation of Example 10.3.3. The R functionrcn.Ris used to generate samples
from a contaminated normal. The following R functionaresimcn.Rcomputes the
simulation and returns the estimate of REn:
aresimcn <- function(n,nsims,eps,vc){
chl <- c(); cxbar <- c()
for(i in 1:nsims){
x <- rcn(n,eps,vc)
chl <- c(chl,wilcox.test(x,conf.int=T)$est)
cxbar <- c(cxbar,t.test(x,conf.int=T)$est)
}
aresimcn <- mses(cxbar,0)/mses(chl,0)
return(aresimcn)}
The functionmses.Rcomputes the MSEs, (10.3.36). All three functions are at the
site listed in the Preface.
For a specific situation setn= 30 with samples generated from the contami-
nated normal distribution with rate of contamination =0.25 and the standard
deviation ratioσc= 3. From Table 10.3.2, the asymptotic ARE is 1.616. Our run
of the functionaresimcn.Rusing 10,000 simulations at these settings produced the
estimate 1.561 for the relative efficiency at sample sizen= 30. This is close to the
asymptotic value. The actual call wasaresimcn(30,10000,.25,3).Wealsoran
the situation with =0.20 andσc= 25. In this case, the estimated RE for samples
of sizen= 30 was 40.934; i.e., we estimate that the Hodges–Lehmann estimator is
41% more efficient that the sample mean at this contaminated normal distribution
for a sample size of 30.
EXERCISES
10.3.1. (a) Forn= 3, expand the mgf (10.3.6) to show that the distribution of
the signed-rank Wilcoxon is given by
j − 6 − 4 − 20246
P(T=j)^18181828181818(b)Obtain the distribution of the signed-rank Wilcoxon forn=4.