642 Nonparametric and Robust StatisticsDefinition 10.9.1.The estimatorθ̂is said to berobustif|IF(x;θ̂)|is bounded
for allx.Hampel (1974) proposed the influence function and discussed its important prop-
erties, a few of which we list below. First, however, we determine the influence
functions of the sample mean and median.
For the sample mean, recall Section 3.4.1 on mixture distributions. The function
Fx,
(t) is the cdf of the random variableU=I 1 − X+[1−I 1 − ]W,whereX,I 1 − ,
andWare independent random variables,Xhas cdfFX(t),W has cdf Δx(t), and
I 1 − isb(1, 1 − ). Hence
E(U)=(1− )E(X)+ E(W)=(1− )E(X)+ x.Denote the mean functional byTμ(FX)=E(X). In terms ofTμ(F), we have just
shown that
Tμ(Fx,
)=(1− )Tμ(FX)+ x.
Therefore,
∂Tμ(Fx,
)
∂
=−Tμ(F)+x.
Hence the influence function of the sample mean is
IF(x;X)=x−μ, (10.9.16)whereμ=E(X). The influence function of the sample mean is linear inxand,
hence, is an unbounded function ofx. Therefore, the sample mean is not a robust
estimator. Another way to derive the influence function is to differentiate implicitly
equation (10.9.10) when this equation is defined forFx,
(t); see Exercise 10.9.6.
Example 10.9.1(Influence Function of the Sample Median).In this example, we
derive the influence function of the sample median,θ̂L 1. In this case, the functional
isTθ(F)=F−^1 (1/2), i.e., the median ofF. To determine the influence function, we
first need to determine the functional at the contaminated cdfFx,
(t), i.e., determine
Fx,
−^1 (1/2). As shown in Exercise 10.9.8, the inverse of the cdfFx,
(t)isgivenby
Fx,
−^1 (u)=⎧
⎨
⎩F−^1(
u
1 −)
u<F(x)
F−^1(
u−
1 −)
u≥F(x),(10.9.17)for 0<u<1. Hence, lettingu=1/2, we get
Tθ(Fx,
)=Fx,
−^1 (1/2) =⎧
⎨
⎩FX−^1(
1 / 2
1 −)
FX−^1( 1
2)
<x
FX−^1(
(1/2)−
1 −)
FX−^1( 1
2)
>x.(10.9.18)Based on (10.9.18) the partial derivative ofFx,
−^1 (1/2) with respect to is seen to be
∂Tθ(Fx,
)
∂ =⎧
⎨
⎩(1/2)(1− )−^2
fX[FX−^1 ((1/2)/(1− ))] F− 1
X( 1
2)
<x
(− 1 /2)(1− )−^2
fX[FX−^1 ({(1/2)− }/{ 1 − })] F− 1
X( 1
2)
>x.(10.9.19)