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

Lin Liu; Chang Li

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arXiv · 2302.08097

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

Lin Liu·Chang Li

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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).

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Lin Liu, Chang Li. 2023-02-16. New $\sqrt{n}$-consistent, numerically stable higher-order influence function estimators. https://arxiv.org/abs/2302.08097

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