Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals

TL;DR

This paper extends the theory of higher order influence functions for complex inference tasks in semi- and non-parametric models, providing new derivations, estimators, and proofs for various functionals and applications.

Contribution

It introduces methods to derive higher order influence functions for products of functionals and applies these to missing data and causal inference scenarios.

Findings

Derived higher order influence functions for product functionals

Developed estimators for mean responses with missing data

Applied influence functions to causal effect estimation with time-varying confounding

Abstract

Robins et al, 2008, published a theory of higher order influence functions for inference in semi- and non-parametric models. This paper is a comprehensive manuscript from which Robins et al, was drawn. The current paper includes many results and proofs that were not included in Robins et al due to space limitation. Particular results contained in the present paper that were not reported in Robins et al include the following. Given a set of functionals and their corresponding higher order influence functions, we show how to derive the higher order influence function of their product. We apply this result to obtain higher order influence functions and associated estimators for the mean of a response Y subject to monotone missingness under missing at random. These results also apply to estimating the causal effect of a time dependent treatment on an outcome Y in the presence of time-varying confounding. Finally, we include an appendix that contains proofs for all theorems that were stated without proof in Robins et al, 2008. The initial part of the paper is closely related to Robins et al, the latter parts differ.