648 Nonparametric and Robust Statisticsx 0 (y 0 −βx 0 ). Therefore, solving for the partial∂T /∂ and evaluating at =0,we
see that the influence function of the LS estimator is given by
IF(x 0 ,y 0 ;β̂LS)=(y 0 −βx 0 )x 0
E(X^2 ). (10.9.45)
Note that the influence function is unbounded in both theY-andX-spaces. Hence
the LS estimator is unduly sensitive to outliers in both spaces. It is not robust.
Based on expression (10.9.40), the Wilcoxon functional at the contaminated
distribution satisfies the equation
0=(1− )∫∞−∞∫∞−∞xφW[F(y−xT )]h(x, y)dxdy+∫∞−∞∫∞−∞xφW[F(y−xT )]dΔ(x 0 ,y 0 )(x, y) (10.9.46)[technically, the cdfFshould be replaced by the actual cdf of the residual, but the
result is the same; see page 477 of Hettmansperger and McKean (2011)]. Proceeding
to implicitly differentiate this expression with respect to ,weobtain0=−∫∞−∞∫∞−∞xφW[F(y−xT )]h(x, y)dxdy+(1− )∫∞−∞∫∞−∞xφ′W[F(y−T x)]f(y−T x)(−x)∂T
∂ h(x, y)dxdy+∫∞−∞∫∞−∞xφW[F(y−xT )]dΔ(x 0 ,y 0 )(x, y)+ B, (10.9.47)where the expression forBis not needed since we are evaluating this partial at
=0. When =0,thenY−TX=eand the random variableseandXare
independent. Hence, upon setting = 0, expression (10.9.47) simplifies to
0=−E[φ′W(F(e))f(e)]E(X^2 )∂T
∂ ∣
∣
∣
∣
=0+φW[F(y 0 −x 0 β)]x 0. (10.9.48)Sinceφ′(u)=
√
12, we finally obtain, as the influence function of the Wilcoxon
estimator,
IF(x 0 ,y 0 ;β̂W)=τφW[F(y 0 −βx 0 )]x 0
E(X^2 ), (10.9.49)whereτ=1/[
√
12∫
f^2 (e)de]. Note that the influence function is bounded in the
Y-space, but it is unbounded in theX-space. Thus, unlike the LS estimator, the
Wilcoxon estimator is robust against outliers in theY-space, but like the LS esti-
mator, it is sensitive to outliers in theX-space. Weighted versions of the Wilcoxon
estimator, though, have bounded influence in both theY-andX-spaces; see the dis-
cussion of the HBR estimator in Chapter 3 of Hettmansperger and McKean (2011).
Exercises 10.9.18 and 10.9.19 asks for derivations, respectively, of the asymptotic
distributions of the LS and Wilcoxon estimators, using their influence functions.