10.9. Robust Concepts 647To obtain the functional corresponding to the Wilcoxon estimate, recall the
association between the ranks and the empirical cdf; see (10.5.14). For Wilcoxon
scores, we have
aW(R(Yi−xciβ)) =φW[
n
n+1
Fn(Yi−xciβ)]. (10.9.39)
Based on the Wilcoxon estimating equations, (10.9.34), and expression (10.9.39),
the functionalTW(H) corresponding to the Wilcoxon estimate satisfies the equation
∫∞
−∞∫∞−∞φW{F[y−TW(H)x]}xh(x, y)dxdy=0. (10.9.40)We next derive the influence functions of the LS and Wilcoxon estimators of
β. In regression models, we are concerned about the influence of outliers in both
theY-andX-spaces. Consider then a point-mass distribution with all its mass at
the point (x 0 ,y 0 ), and let Δ(x 0 ,y 0 )(x, y) denote the corresponding cdf. Let denote
the probability of sampling from this contaminating distribution, where 0< <1.
Hence, consider the contaminated distribution with cdf
H (x, y)=(1− )H(x, y)+ Δ(x 0 ,y 0 )(x, y). (10.9.41)Because the differential is a linear operator, we have
dH (x, y)=(1− )dH(x, y)+ dΔ(x 0 ,y 0 )(x, y), (10.9.42)wheredH(x, y)=h(x, y)dxdy;thatis,dcorresponds to the second mixed partial
∂^2 /∂x ∂y.
By (10.9.38), the LS functionalT at the cdfH (x, y) satisfies the equation
0=(1− )∫∞−∞∫∞−∞x(y−xT )h(x, y)dxdy+∫∞−∞∫∞−∞x(y−xT )dΔ(x 0 ,y 0 )(x, y).
(10.9.43)
To find the partial derivative ofT with respect to , we simply implicitly differen-
tiate expression (10.9.43) with respect to , which yields
0=−∫∞−∞∫∞−∞x(y−T x)h(x, y)dxdy+(1− )∫∞−∞∫∞−∞x(−x)∂T
∂ h(x, y)dxdy+∫∞−∞∫∞−∞x(y−xT )dΔ(x 0 ,y 0 )(x, y)+ B, (10.9.44)where the expression forBis not needed since we are evaluating this partial at
=0. Noticethatat =0,y−T x=y−Tx=y−βx. Hence, at = 0, the first
expression on the right side of (10.9.44) is 0, while the second expression becomes
−E(X^2 )(∂T/∂ ), where the partial is evaluated at 0. Finally, the third expression
is the expected value of the point-mass distribution Δ(x 0 ,y 0 ),whichis,ofcourse,