Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

646 Nonparametric and Robust Statistics

estimator is

β̂

LS=

∑n

∑i=1xciYi

n

i=1x

2

ci

, (10.9.33)

which agrees with expression (9.6.5) of Chapter 9. The geometry of the LS estimator

is discussed in Remark 9.6.2.

For our second procedure, we consider the estimate of slope discussed in Section

10.7. This is a rank-based estimate based on an arbitrary score function. In this

section, we restrict our discussion to the linear (Wilcoxon) scores; i.e., the score

function is given byφW(u)=

√

12[u−(1/2)], where the subscriptWdenotes the

Wilcoxon score function. The estimating equation of the rank-based estimator ofβ

is given by expression (10.7.8), which for the Wilcoxon score function is

∑n

i=1

aW(R(Yi−xciβ))xci=0, (10.9.34)

whereaW(i)=φW[i/(n+1)]. This equation is the analog of the LS normal equation.
See Exercise 10.9.12 for a geometric interpretation.

Influence Functions

To determine the robustness properties of these procedures, first consider a prob-
ability model corresponding to Model (10.9.31), in whichX, in addition toY,is
a random variable. Assume that the random vector (X, Y) has joint cdf and pdf,
H(x, y)andh(x, y), respectively, and satisfies

Y=βX+e, (10.9.35)

where the random variableehas cdf and pdfF(t)andf(t), respectively, andeand
Xare independent. Since we have centered thexs, we also assume thatE(X)=0.
As Exercise 10.9.13 shows,

P(Y≤t|X=x)=F(t−βx), (10.9.36)

and, hence,YandXare independent if and only ifβ=0.
The functional for the LS estimator easily follows from the LS normal equation
(10.9.32). LetHndenote the empirical cdf of the pairs (x 1 ,y 1 ),(x 2 ,y 2 ),...,(xn,yn);
that is,Hnis the cdf corresponding to the discrete distribution, which puts probabil-
ity (mass) of 1/non each point (xi,yi). Then the LS estimating equation, (10.9.32),
can be expressed as an expectation with respect to this distribution as
∑n

i=1

(yi−xciβ)xci

1

n

=0. (10.9.37)

For the probability model, (10.9.35), it follows that the functionalTLS(H) corre-
sponding to the LS estimate is the solution to the equation
∫∞

−∞

∫∞

−∞

[y−TLS(H)x]xh(x, y)dxdy=0. (10.9.38)