Tutorial: Deriving The Efficient Influence Curve for Large Models

Abstract

This paper aims to provide a tutorial for upper level undergraduate and graduate students in statistics, biostatistics and epidemiology on deriving influence functions for non-parametric and semi-parametric models. The author will build on previously known efficiency theory and provide a useful identity and formulaic technique only relying on the basics of integration which, are self-contained in this tutorial and can be used in most any setting one might encounter in practice. The paper provides many examples of such derivations for well-known influence functions as well as for new parameters of interest. The influence function remains a central object for constructing efficient estimators for large models, such as the one-step estimator and the targeted maximum likelihood estimator. We will not touch upon these estimators at all but readers familiar with these estimators might find this tutorial of particular use.