Theory of Statistics - George Mason University

p(y)
(1-ε)p(y)
The Influence Function
x1
ε
q
yπ
p(y)
(1-ε)p(y)
Figure 8.5. Quantile of the ɛ-Mixture Distribution
8.7 Robust Inference 601
The extent of the perturbation depends on ɛ, and so we are interested in the
relative effect; in particular, the relative effect as ɛ approaches zero.
The influence function for the functional Υ and the CDF P, defined at x
as
Υ(Px,ɛ) − Υ(P)
φΥ,P(x) = lim
ɛ↓0 ɛ
yπ
q
x2
ε
(8.82)
if the limit exists, is a measure of the sensitivity of the distributional measure
defined by Υ to a perturbation of the distribution at the point x. The influence
function is also called the influence curve, and denoted by IC.
The limit in equation (8.82) is the right-hand Gâteaux derivative of the
functional Υ at P and x.
The influence function can also be expressed as the limit of the derivative
of Υ(Px,ɛ) with respect to ɛ:
φΥ,P(x) = lim
ɛ↓0
∂
∂ɛ Υ(Px,ɛ). (8.83)
This form is often more convenient for evaluating the influence function.
Some influence functions are easy to work out, for example, the influence
function for the functional M in equation (8.79) that defines the mean of a
distribution, which we denote by µ. The influence function for this functional
operating on the CDF P at x is
Theory of Statistics c○2000–2013 James E. Gentle