10.3. Signed-Rank Wilcoxon 59710.3.2.Consider the location Model (10.3.35). Assume that the pdf of the random
errors,f(x), is symmetric about 0. Letθ̂be a location estimator ofθ. Assume that
E(θ̂^4 ) exists.(a)Show thatθ̂is an unbiased estimator ofθ.
Hint: Assume without loss of generality thatθ = 0; start withE(θ̂)=
E[θ̂(X 1 ,...,Xn)]; and use the fact thatXiis symmetrically distributed about
0.(b)As in Section 10.3.4, suppose we generatensindependent samples of sizen
from the pdff(x) which is symmetric about 0. For theith sample, let̂θibe
the estimate ofθ. Show thatn−s^1∑ns
i=1
̂θi^2 →V(θ̂), in probability.10.3.3.Modify the code of the R functionaresimcn.Rso it samples from the
N(0,1) distribution. Estimate the RE between the Hodges–Lehmann estimator
andXfor the sample sizesn=15, 25 ,50 and 100. Use 10,000 simulations for each
sample size. Compare your results to the asymptotic ARE which is 0.955.
10.3.4.Consider the self rival data presented in Exercise 4.6.5. Recall that it is
a paired design consisting of the pairs (Selfi,Rivali), fori=1,...,20, where Selfi
and Rivaliare the running times for circling the bases for the respective treat-
ments of Self motivation and Rival motivation. The data can be found in the file
selfrival.rda.LetXi=Selfi−Rivalidenote the paired differences and model
these in the location model asXi=θ+ei. Consider the hypothesesH 0 :θ=0
versusH 1 :θ =0.
(a)Obtain the signed-rank test statistic andp-value for these hypotheses. State
the conclusion (in terms of the data) using the level 0.05.(b)Obtain thettest statistic andp-value and conclude using the level 0.05.
(c)To see the effect that an outlier has on these two analyses, change the 20th
rival time from 17.88 to 178.8. Comment on how the analyses changed due to
the outlier.(d)Obtain 95% confidence intervals forθfor both analyses for the original data
and the changed data. Comment on how the confidence intervals changed due
to the outlier.
10.3.5.Assume thatf(x) has the contaminated normal pdf given in expression
(3.4.17). Derive expression (10.3.30) and use it to obtain ARE(W, t)forthispdf.
10.3.6.Use the asymptotic null distribution ofT+, (10.3.10), to obtain the ap-
proximation (10.3.34) tocW 1.10.3.7.For a vectorv∈Rn, define the function
‖v‖=∑ni=1R(|vi|)|vi|. (10.3.38)Show that this function is a norm onRn; that is, it satisfies the properties