# Stabilized Higher-Order Influence Functions: Statistical Theory of a Class of Bilinear Forms

arXiv: https://arxiv.org/abs/2607.04743 · HTML: https://mlsanity.com/paper/2607.04743/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 68 references that are in the map.
- First posted on arXiv 6 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Doubly-robust inference and optimality in structure-agnostic models with smoothness](https://mlsanity.com/paper/2405.08525/) (2024)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Method-of-Moments Inference for GLMs and Doubly Robust Functionals under Proportional Asymptotics](https://mlsanity.com/paper/2408.06103/) (2024)
- [On the Asymptotic Inadmissibility of Double Machine Learning Estimators Under Structure-Agnostic Models](https://mlsanity.com/paper/2606.22391/) (2026)
- [When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators](https://mlsanity.com/paper/2411.02909/) (2024)
- [Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals](https://mlsanity.com/paper/1601.05820/) (2016)
- [New √n-consistent, numerically stable higher-order influence function estimators](https://mlsanity.com/paper/2302.08097/) (2023)
- [Root-n consistent semiparametric learning with high-dimensional nuisance functions under minimal sparsity](https://mlsanity.com/paper/2305.04174/) (2023)

## Abstract (arXiv)

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 Omega 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 Omega 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.