Influence Function: Local Robustness and Efficiency

TL;DR

This paper presents a unified, differentiation-based framework for deriving influence functions applicable to various models, enhancing understanding of robustness and efficiency in semiparametric estimation.

Contribution

It introduces a general approach to derive influence functions across model types, extending to infinite-dimensional parameters and clarifying conditions for efficiency equivalence.

Findings

Unified influence function derivation for parametric and nonparametric models

Automatic generation of locally robust moment functions in semiparametric estimation

Conditions established for efficiency equivalence in joint versus plug-in estimation

Abstract

This paper introduces a direct differentiation-based framework that unifies the derivation of influence functions across parametric, nonparametric, and semiparametric models. We show that the Riesz representer of the functional derivative is obtained by orthogonally projecting the identification function onto the subspace of mean-zero functions. Consequently, the influence function emerges as a linear transformation of this centered moment function. The approach extends seamlessly to infinite-dimensional parameters, revealing a common algebraic form for influence functions across both finite- and infinite-dimensional parameters. Applied to semiparametric multi-step plug-in estimation, our method automatically yields locally robust moment functions and provides an explicit closed-form expression for the adjustment term. Finally, we leverage this framework to revisit the joint versus plug-in estimation debate, establishing verifiable sufficient conditions for their semiparametric efficiency equivalence even when nuisance parameters are over-identified.