Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms
Abstract
Higher-order influence functions, introduced in a series of articles (Robins et al., 2008, 2009a; van der Vaart, 2014; Robins et al., 2016, 2023; Liu et al., 2017), are a unified framework for constructing rate-optimal point estimates of a class of statistical functionals under various complexity-reducing assumptions on the posited statistical model that generates the observed data. Although higher-order (influence functions) estimators are theoretically appealing, they have very limited practical uptake compared to their first-order counterparts. The original higher-order estimators proposed in Robins et al. (2008) and Robins et al. (2017) involve nonparametric density estimation of multi-dimensional covariates, a highly nontrivial statistical and computational problem on its own. The density estimator is, in turn, used in the evaluation of the inverse population Gram matrix Ω of a set of k-dimensional basis transformations of covariates. There, k is allowed to be as large as o (n2). To partially address this potential shortcoming, Liu et al. (2017) restrict k to $o (n)$ and instead estimate Ω directly using the inverse sample Gram matrix estimator, but computed from an independent sample often obtained by sample-splitting. Liu et al. (2017) refer to this alternative estimator as the empirical higher-order estimator. Although the empirical higher-order estimator bypasses density estimation, it suffers from numerical instability due to inverting a large-dimensional sample Gram matrix. In this article, for a class of bilinear forms/functionals that often appear in substantive fields, we propose a new stabilized higher-order estimator without sample splitting, which exhibits more stable finite-sample performance compared to the empirical higher-order estimator. We also prove that this new class of higher-order estimators enjoys similar statistical guarantees.
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BibTeX
@misc{liu2026stabilized,
title = {{Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms}},
author = {Na Liu and Chang Li and Yujia Gu and Lin Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.04743},
eprint = {2607.04743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04743},
}
RIS
TY - UNPB
TI - Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms
AU - Na Liu
AU - Chang Li
AU - Yujia Gu
AU - Lin Liu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2607.04743
UR - https://arxiv.org/abs/2607.04743
AB - Higher-order influence functions, introduced in a series of articles (Robins et al., 2008, 2009a; van der Vaart, 2014; Robins et al., 2016, 2023; Liu et al., 2017), are a unified framework for constructing rate-optimal point estimates of a class of statistical functionals under various complexity-reducing assumptions on the posited statistical model that generates the observed data. Although higher-order (influence functions) estimators are theoretically appealing, they have very limited practical uptake compared to their first-order counterparts. The original higher-order estimators proposed in Robins et al. (2008) and Robins et al. (2017) involve nonparametric density estimation of multi-dimensional covariates, a highly nontrivial statistical and computational problem on its own. The density estimator is, in turn, used in the evaluation of the inverse population Gram matrix Ω of a set of k-dimensional basis transformations of covariates. There, k is allowed to be as large as o (n^2). To partially address this potential shortcoming, Liu et al. (2017) restrict k to $o (n)$ and instead estimate Ω directly using the inverse sample Gram matrix estimator, but computed from an independent sample often obtained by sample-splitting. Liu et al. (2017) refer to this alternative estimator as the empirical higher-order estimator. Although the empirical higher-order estimator bypasses density estimation, it suffers from numerical instability due to inverting a large-dimensional sample Gram matrix. In this article, for a class of bilinear forms/functionals that often appear in substantive fields, we propose a new stabilized higher-order estimator without sample splitting, which exhibits more stable finite-sample performance compared to the empirical higher-order estimator. We also prove that this new class of higher-order estimators enjoys similar statistical guarantees.
ER -