Foundations of TMLE

Abstract

An estimator of a parameter is a mapping from the data set to the parameter space. Estimators that are empirical means of a function of the unit data structure are asymptotically consistent and normally distributed due to the CLT. Such estimators are called linear in the empirical probability distribution. Most estimators are not linear, but many are approximately linear in the sense that they are linear up to a negligible (in probability) remainder term. One states that the estimator is asymptotically linear, and the relevant function of the unit data structure, centered to have mean zero, is called the influence curve of the estimator. How does one prove that an estimator is asymptotically linear? One key step is to realize that an estimator is a mapping from a possibly very large collection of empirical means of functions of the unit data structure into the parameter space. Such a collection of empirical means is called an empirical process whose behavior with respect to uniform consistency and the uniform CLT is established in empirical process theory. In this section we present succinctly that (1) a uniform central limit theorem for the vector of empirical means, combined with (2) differentiability of the estimator as a mapping from the vector of empirical means into the parameter space yields the desired asymptotic linearity. This method for establishing the asymptotic linearity and normality of the estimator is called the functional delta method (van der Vaart and Wellner 1996; Gill 1989).