## Influence function

One of the useful tools in asymptotic and robust statistics often not discussed in econometric textbooks (even the advanced ones) is the influence function. In this post, I try to explain the concept and its applications.

Perhaps a good starting point is Glivenko–Cantelli theorem. According to this theorem under very general conditions $\widehat{F} \rightarrow F$ uniformly where “F” is the CDF and $\widehat{F}$ is the empirical-CDF. Many estimators (mean, median, quantile, Lorenz, Gini, etc) are functions of the CDF. A fundamental question in nonparametric analysis is this: under what condition $T(\widehat{F})\rightarrow T(F)$ where “T” is some function? The answer is that we have this result if the functional “T” is (Hadamard) differentiable with respect to function “F“.

How the derivative with respect to a function e.g.”F” is defined? One definition is Gateaux derivative which is an extension of the directional derivative. The Gateaux derivative of functional “T at function “F” in the direction of function “G” is defined by

$D_{F}(T,G)=lim_{\varepsilon \rightarrow 0}\dfrac{T[(1-\varepsilon)F+\varepsilon G]-T(F) }{\varepsilon}$

Influence function is a Gateaux derivative where $G=\delta _{x}$ and $\delta _{x}$ is point mass function at “x” or Dirac’s delta function i.e. it “measures the change in “T” if an infinitesimally small part of F is replaced by a pointmass at x”. Influence function has several applications, two of them are:

• Robustness: An estimator is robust if it has a bounded influence function and is not robust otherwise. For example, mean is not a robust estimator but median is.
• Deriving asymptotic variance: it has also been shown that under certain regularity conditions

$\widehat{T}\rightarrow T(F)+E_{n}(IF)$

$\Rightarrow$ $Var(\sqrt[]{n}\widehat{T})\rightarrow Var{(IF)}$

i.e. to derive asymptotic variance of an estimator it suffices to $\int IF(y)^2dF(y)$

For more on influence function see van der Vaart (2000). For applications in income distribution analysis see Cowell and Flachaire or this. See also this for a recent possibly important contribution.