Theory of Statistics - George Mason University

8.7 Robust Inference 603
Notice that the actual value of x is not in the influence function; only whether
x is less than, equal to, or greater than the quantile. Notice also that, unlike
influence function for the mean, the influence function for a quantile is
bounded; hence, a quantile is less sensitive than the mean to perturbations
of the distribution. Likewise, quantile-based measures of scale and skewness,
as in equations (1.113) and (1.114), are less sensitive than the moment-based
measures to perturbations of the distribution.
The functionals LJ and Mρ defined in equations (1.116) and (1.117), depending
on J or ρ, can also be very insensitive to perturbations of the distribution.
The mean and variance of the influence function at a random point are of
interest; in particular, we may wish to restrict the functional so that
and
8.7.2 Robust Estimators
E(φΥ,P(X)) = 0
E (φΥ,P(X)) 2 < ∞.
If a distributional measure of interest is defined on the CDF as Υ(P), we
are interested in the performance of the plug-in estimator Υ(Pn); specifically,
we are interested in Υ(Pn) − Υ(P). This turns out to depend crucially on
the differentiability of Υ. If we assume Gâteaux differentiability, from equation
(0.1.115), we can write
√ n (Υ(Pn) − Υ(P)) = ΛP( √ n(Pn − P)) + Rn
= 1
√ n
i
φΥ,P(Yi) + Rn
where the remainder Rn → 0.
We are interested in the stochastic convergence. First, we assume
and
E(φΥ,P(X)) = 0
E (φΥ,P(X)) 2 < ∞.
Then the question is the stochastic convergence of Rn. Gâteaux differentiability
does not guarantee that Rn converges fast enough. However, ρ-Hadamard
differentiability, does imply that that Rn is in oP(1), because it implies that
norms of functionals (with or without random arguments) go to 0. We can
also get that Rn is in oP(1) by assuming Υ is ρ-Fréchet differentiable and that
Theory of Statistics c○2000–2013 James E. Gentle