Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

10.9. Robust Concepts 641

The solution to the above equation is, of course,T(FX)=E(X).
Likewise, we can write the estimating equation (EE), (10.9.3), which defines the
sample median, as
∑n

i=1

sgn(Xi−θ)

1

n

=0. (10.9.11)

The corresponding equation for the functionalθ=T(FX) is the solution of the
equation ∫
∞

−∞

sgn[y−T(FX)]fX(y)dy=0. (10.9.12)

Note that this can be written as

0=−

∫T(FX)

−∞

fX(y)dy+

∫∞

T(FX)

fX(y)dy=−FX[T(FX)] + 1−FX[T(FX)].

HenceFX[T(FX)] = 1/2orT(FX)=FX−^1 (1/2). ThusT(FX) is the median of the
distribution ofX.
Now we want to consider how a given functionalT(FX) changes relative to some
perturbation. The analog of adding an outlier toF(t)istoconsiderapoint-mass
contaminationof the cdfFX(t)atapointx.Thatis,for>0, let

Fx,

(t)=(1− )FX(t)+ Δx(t), (10.9.13)

where Δx(t) is the cdf with all its mass atx; i.e.,

Δx(t)=

{

0 t<x

1 t≥x.

(10.9.14)

The cdfFx,
(t) is a mixture of two distributions. When sampling from it, (1− )100%
of the time an observation is drawn fromFX(t), while 100% of the timex(an
outlier) is drawn. Soxhas the flavor of the outlier in the sensitivity curve. As
Exercise 10.9.4 shows,Fx,
(t)isinan neighborhood ofFX(t); that is, for allx,
|Fx,
(t)−FX(t)|≤. Hence the functional atFx,
(t) should also be close toT(FX).
The concept for functionals, corresponding to the sensitivity curve, is the function

IF(x;̂θ) = lim

→ 0

T(Fx,

)−T(FX)

, (10.9.15)

provided the limit exists. The function IF(x;θ̂) is called theinfluence functionof
the estimator̂θatx. As the notation suggests, it can be thought of as a derivative
of the functionalT(Fx
) with respect to evaluated at 0, and we often determine
it this way. Note that for small,

T(Fx,

)≈T(FX)+ IF(x;̂θ);

hence, the change of the functional due to point-mass contamination is approxi-
mately directly proportional to the influence function. We want estimators, whose
influence functions are not sensitive to outliers. Further, as mentioned above, for
anyx,Fx,
(t)isclosetoFX(t). Hence, at least, the influence function should be a
bounded function ofx.