10.9. Robust Concepts 649Breakdown Points
Breakdown for the regression model is based on the corruption of the sample in
Model (10.9.31), that is, the sample (xc 1 ,Y 1 ),...,(xcn,Yn). Based on the influence
functions for both the LS and Wilcoxon estimators, it is clear that corrupting one
xibreaks down both estimators. This is shown in Exercise 10.9.14. Hence the
breakdown point of each estimator is 0. The HBR estimator (weighted version of
the Wilcoxon estimator) has bounded influence in both spaces and can achieve 50%
breakdown; see Chang et al. (1999) and Hettmansperger and McKean (2011).Intercept
In practice, the linear model usually contains an intercept parameter; that is, the
model is given by (10.9.30) with intercept parameterα.Noticethatαis a location
parameter of the random variablesYi−βxci. This suggests an estimate of location
on the residualsYi−βx̂ci. For LS, we take the sample mean of the residuals; i.e.,α̂LS=n−^1∑ni=1(Yi−β̂LSxci)=Y, (10.9.50)because thexcis are centered. For the Wilcoxon fit, several choices seem appropriate.
We use the median of the Wilcoxon residuals. That is, let
α̂W=med 1 ≤i≤n{Yi−β̂Wxci}. (10.9.51)For the Wilcoxon fit of the regression model, computation is discussed in Remark
10.7.1. As there, we recommend the CRAN packageRfitdeveloped by Kloke and
McKean (2014). The R package^1 hbrfitcomputes the high breakdown HBR fit.
EXERCISES10.9.1.Consider the location model as defined in expression (10.9.1). Letθ̂=Argminθ‖X−θ 1 ‖^2
LS,where‖·‖^2 LSis the square of the Euclidean norm. Show that̂θ=x.
10.9.2.Obtain the sensitivity curves for the sample mean, the sample median and
the Hodges–Lehmann estimator for the following data set. Evaluate the curves at
the values−300 to 300 in increments of 10 and graph the curves on the same plot.
Compare the sensitivity curves.
− 95812 − 1 −37 0 11 21
18 − 24 − 4 − 53 −99 8Note that the R commandwilcox.test(x,conf.int=T)$estcomputes the Hodges
Lehmann estimate for the R vectorx.
(^1) Downloadable at https://github.com/kloke/