10.7. Simple Linear Model 627Hence the corresponding estimate ofβis given byβ̂φ, which solves the estimating
equations
Tφ(β̂φ)≈ 0. (10.7.8)Similar to Theorem 10.5.1, it can be shown thatTφ(β) is a decreasing step function
ofβthat steps down at each sample slope (Yj−Yi)/(xj−xi), fori =j.Thusthe
estimate exists. It cannot be obtained in closed form, but simple iterative techniques
can be used to find the solution. In the regression problem, though, prediction ofY
is often of interest, which also requires an estimate ofα. Notice that such an estimate
can be obtained as a location estimate based on residuals. This is discussed in some
detail in Section 3.5.2 of Hettmansperger and McKean (2011). For our purposes,
we consider the median of the residuals; that is, we estimateαas
̂α=med{Yi−β̂φ(xi−x)}. (10.7.9)Remark 10.7.1(Computation).The Wilcoxon estimates of slope and intercept are
computed by several packages. We recommend the CRAN packageRfitdeveloped
by Kloke and McKean (2012). Chapter 4 of the book by Kloke and McKean (2014)
discusses the use ofRfitfor the simple regression model (10.7.1). Rfithas code
for many score functions, including the Wilcoxon scores, normal scores, as well as
scores appropriate for skewed error distributions. The computations in this section
are performed byRfit. Also, the minitab commandrregrobtains the Wilcoxon
fit. Terpstra and McKean (2005) have written a collection of R functions,ww,which
obtains the fit using Wilcoxon scores.
Example 10.7.2(Telephone Data).Consider the regression data discussed in Ex-
ercise 9.6.3. Recall that the responses (y) for this data set are the numbers of
telephone calls (tens of millions) made in Belgium for the years1950–1973, while
time in years serves as the predictor variable (x). The data are plotted in Figure
10.7.1. The data are in the filetelephone.rda. For this example, we use Wilcoxon
scores to fit Model (10.7.1). The code and partial results (including the plot with
overlaid fits) are:
fitls <- lm(numcall~year); fitrb <- rfit(numcall~year)
fitls$coef; fitrb$coef # Result -26.0, 0.504; -7.1, 0.145
plot(numcall~year,xlab="Year",ylab="Number of calls")
abline(fitls); abline(fitrb,lty=2)
legend(50,15,c("LS-Fit","Wilcoxon-Fit"),lty=c(1,2))
Thus, the Wilcoxon fitted value isŶφ,i=− 7 .1+0. 145 xiwhichisplottedinFigure10.7.1. The least squares fitŶLS,i=− 26 .0+0. 504 xi, is also plotted. Note that the
Wilcoxon fit is much less sensitive to the outliers than the least squares fit.
The outliers in this data set were recording errors; see page 25 of Rousseeuw
and Leroy (1987) for more discussion.
Similar to Lemma 10.2.1, a translation property holds for the processT(β)given
by
Eβ[T(0)] =E 0 [T(−β)]; (10.7.10)