640 Nonparametric and Robust StatisticsIn contrast, consider the sample median in which the sample sizenis odd. In
this case, the sample median iŝθL 1 ,n =x(r),wherer =(n+1)/2. When the
additional pointxis added, the sample size becomes even and the sample median
θ̂L 1 ,n+1is the average of the middle two order statistics. Ifxvaries between these
two order statistics, then there is some change between thêθL 1 ,nandθ̂L 1 ,n+1.But
oncexmoves beyond these middle two order statistics, there is no change. Hence
S(x;θ̂L 1 ,n) is a bounded function ofx. Therefore,̂θL 1 ,nis much less sensitive to an
outlier than the sample mean.
Because the Hodges–Lehmann estimator̂θHL, (10.9.4), is also a median, its
sensitivity curve is also bounded. Exercise 10.9.2 provides a numerical illustration
of these sensitivity curves.
Influence FunctionsOne problem with the sensitivity curve is its dependence on the sample. In earlier
chapters, we compared estimators in terms of their variances which are functions of
the underlying distribution. This is the type of comparison we want to make here.
Recall that the location model (10.9.1) is the model of interest, whereFX(t)=
F(t−θ)isthecdfofXandF(t)isthecdfofε. As discussed in Section 10.1, the
parameterθis a function of the cdfFX(x). It is convenient, then, to use functional
notationθ=T(FX), as in Section 10.1. For example, ifθis the mean, thenT(FX)
is defined as
T(FX)=∫∞−∞xdFX(x)=∫∞−∞xfX(x)dx, (10.9.7)while ifθis the median, thenT(FX) is defined as
T(FX)=FX−^1(
1
2). (10.9.8)
It was shown in Section 10.1 that for a location functional,T(FX)=T(F)+θ.
Estimating equations (EE) such as those defined in expressions (10.9.2) and
(10.9.3) are often quite intuitive, for example, based on likelihood equations or
methods such as least squares. On the other hand, functionals are more of an ab-
stract concept. But often the estimating equations naturally lead to the functionals.
We outline this next for the mean and median functionals.
LetFnbe the empirical distribution function of the realized samplex 1 ,x 2 ,...,xn.
That is,Fnis the cdf of the distribution which puts massn−^1 on eachxi; see (10.1.1).
Note that we can write the estimating equation (10.9.2), which defines the sample
mean as n
∑
i=1(xi−θ)1
n=0. (10.9.9)This is an expectation using the empirical distribution. SinceFn→FXin proba-
bility, it would seem that this expectation converges to
∫∞−∞[x−T(FX)]fX(x)dx=0. (10.9.10)