652 Nonparametric and Robust Statistics(b)‖cv‖W=|c|‖v‖W, for allc∈R.(c)‖v+w‖W≤‖v‖W+‖w‖W,for allv,w∈Rn.
Hint:Determine the permutations, say,ikandjkof the integers{ 1 , 2 ,...,n},
which maximize∑n
k=1cikdjkfor the two sets of numbers{c^1 ,...,cn}and
{d 1 ,...,dn}.10.9.12.Remark 9.6.2 discusses the geometry of the LS estimate ofβ.Thereisan
analogous geometry for the Wilcoxon estimate. Using the norm‖·‖Wdefined in
expression (10.9.53) of the last exercise, let
β̂∗=Argmin‖Y−Xcβ‖W,whereY′=(Y 1 ,...,Yn)andX′c=(xc 1 ,...,xcn). Thusβ̂∗minimizes the distance
betweenYand the space spanned by the vectorXc.
(a)Using expression (10.9.54), show thatβ̂∗ satisfies the Wilcoxon estimating
equation (10.9.34). That is,β̂∗=β̂W.(b)LetŶW=Xcβ̂W andY−ŶWdenote the Wilcoxon vectors of fitted values
and residuals, respectively. Sketch a figure analogous to the LS Figure 9.6.3
but with these vectors on it. Note that your figure may not contain a right
angle.(c)For the Wilcoxon regression procedure, determine a vector (not 0 )thatis
orthogonal toŶW.10.9.13.For Model (10.9.35), show that equation (10.9.36) holds. Then show that
YandXare independent if and only ifβ= 0. Hence independence is based on the
value of a parameter. This is a case where normality is not necessary to have this
independence property.
10.9.14.Consider the telephone data discussed in Example 10.7.2 and given in the
rda-filetelephone.rda. It is easily seen in Figure 10.7.1 that there are seven outliers
in theY–space. Based on the estimates discussed in this example, the Wilcoxon
estimate of slope is robust to these outliers, while the LS estimate is highly sensitive
to them.
(a)For this data set, change the last value ofxfrom 73 to 173. Notice the drastic
change in the LS fit.(b)Obtain the Wilcoxon estimate for the changed data in part (a). Notice that
it has a drastic change also. To obtain the Wilcoxon fit, see Remark 10.7.1
on computation.(c)Using the Wilcoxon estimates of Example 10.7.2, change the the value ofY
atx= 173 to the predicted value ofY basedontheWilcoxonestimatesof
Example 10.7.2. Note that this point is a “good” point at the outlyingx;
that is, it fits the model. Now determine the Wilcoxon and LS estimates.
Comment on them.