10.9. Robust Concepts 643Evaluating this partial derivative at = 0, we arrive at the influence function of the
median:
IF(x;̂θL 1 )={
1
2 fX(θ) θ<x
− 1
2 fX(θ) θ>x}
=
sgn(x−θ)
2 f(θ), (10.9.20)whereθis the median ofFX. Because this influence function is bounded, the sample
median is a robust estimator.
As derived on p. 46 of Hettmansperger and McKean (2011), the influence func-tion of the Hodges–Lehmann estimator,̂θHL,atthepointxis given by:
IF(x;̂θHL)=FX(x)− 1 / 2
∫∞
−∞f2
X(t)dt. (10.9.21)
Since a cdf is bounded, the Hodges–Lehmann estimator is robust.
We now list three useful properties of the influence function of an estimator.
Note that for the sample mean,E[IF(X;X)] =E[X]−μ=0. Thisistruein
general. Let IF(x)=IF(x;̂θ) denote the influence function of the estimator̂θwith
functionalθ=T(FX). Then
E[IF(X)] = 0, (10.9.22)
provided expectations exist; see Huber (1981) for a discussion. Hence, for the second
property, we have
Var[IF(X)] =E[IF^2 (X)], (10.9.23)
provided the squared expectation exists. A third property of the influence function
is the asymptotic result
√
n[̂θ−θ]=1
√
n∑ni=1IF(Xi)+op(1). (10.9.24)Assume that the variance (10.9.23) exists, then because IF(X 1 ),...,IF(Xn) are iid
with finite variance, the simple Central Limit Theorem and (10.9.24) imply that
√
n[̂θ−θ]
D
→N(0,E[IF^2 (X)]). (10.9.25)Thus we can obtain the asymptotic distribution of the estimator from its influence
function. Under general conditions, expression (10.9.24) holds, but often the verifi-
cation of the conditions is difficult and the asymptotic distribution can be obtained
more easily in another way; see Huber (1981) for a discussion. In this chapter,
though, we use (10.9.24) to obtain asymptotic distributions of estimators. Suppose
(10.9.24) holds for the estimatorŝθ 1 andθ̂ 2 , which are both estimators of the same
functional, say,θ. Then, letting IFidenote the influence function ofθ̂i,i=1,2, we
can express the asymptotic relative efficiency between the two estimators as
ARE(θ̂ 1 ,̂θ 2 )=E[IF^22 (X)]
E[IF^21 (X)]. (10.9.26)
As an example, we consider the sample median.