New $\sqrt{n}$-consistent, numerically stable higher-order influence function estimators

TL;DR

This paper introduces numerically stable higher-order influence function estimators that are both theoretically sound and practically useful for estimating complex statistical functionals across various fields.

Contribution

It develops a new class of numerically stable HOIF estimators with proven statistical, numerical, and computational guarantees, advancing their practical applicability.

Findings

New stable HOIF estimators up to second order

Proven statistical and computational guarantees

Enhanced practical utility of influence function methods

Abstract

Higher-Order Influence Functions (HOIFs) provide a unified theory for constructing rate-optimal estimators for a large class of low-dimensional (smooth) statistical functionals/parameters (and sometimes even infinite-dimensional functions) that arise in substantive fields including epidemiology, economics, and the social sciences. Since the introduction of HOIFs by Robins et al. (2008), they have been viewed mostly as a theoretical benchmark rather than a useful tool for statistical practice. Works aimed to flip the script are scant, but a few recent papers Liu et al. (2017, 2021b) make some partial progress. In this paper, we take a fresh attempt at achieving this goal by constructing new, numerically stable HOIF estimators (or sHOIF estimators for short with ``s'' standing for ``stable'') with provable statistical, numerical, and computational guarantees. This new class of sHOIF estimators (up to the 2nd order) was foreshadowed in synthetic experiments conducted by Liu et al. (2020a).