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

arXiv:2607.04743 · math.ST, econ.EM, stat.ME, stat.ML, stat.TH · Submitted 2026-07-06 · Read on arXiv

math.ST, econ.EM, stat.ME, stat.ML, stat.TH

Submitted: 2026-07-06

Updated: 2026-09-20

Comments: This paper improves and supersedes a previous draft by the second and last authors: arXiv:2302.08097

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Terminology

• Influence Functions — 11× in this paper · explained in 3 episodes
• Causal Inference — 4× in this paper · explained in 5 episodes
• Conditional Independence — 3× in this paper · explained in 2 episodes
• Theoretical Guarantees — 3× in this paper · explained in 2 episodes
• Consistency — 2× in this paper · explained in 3 episodes
• Unified Framework — 2× in this paper · explained in 6 episodes
• Annual Symposium on Foundations of Computer Science — 1× in this paper
• Asymptotic Normality — 1× in this paper · explained in 2 episodes
• Bayesian Inference — 1× in this paper · explained in 6 episodes
• Benchmark — 1× in this paper · explained in 2 episodes
• Effective Rank — 1× in this paper · explained in 2 episodes
• Embeddings — 1× in this paper · explained in 2 episodes
• Exchangeability — 1× in this paper · explained in 2 episodes
• Fisher Information — 1× in this paper · explained in 4 episodes
• Heteroscedasticity — 1× in this paper · explained in 2 episodes
• Isaac Newton Institute — 1× in this paper
• Neyman orthogonality — 1× in this paper · explained in 2 episodes
• Regularization — 1× in this paper · explained in 3 episodes
• Robustness — 1× in this paper · explained in 6 episodes
• Sample Complexity — 1× in this paper · explained in 20 episodes
• Score Function — 1× in this paper · explained in 4 episodes
• Sparsity — 1× in this paper · explained in 5 episodes

Sources

• Fisher-type information involving higher order derivatives
• Higher-Order Neyman Orthogonality in Moment-Condition Models
• Fast convergence rates for dose-response estimation
• Doubly-robust inference and optimality in structure-agnostic models with smoothness
• Tail Bounds for Canonical $U$-Statistics and $U$-Processes with Unbounded Kernels
• Method-of-Moments Inference for GLMs and Doubly Robust Functionals under Proportional Asymptotics
• On computing and the complexity of computing higher-order U-statistics, exactly
• When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators
• New $\sqrt{n}$-consistent, numerically stable higher-order influence function estimators
• Semiparametric Efficient Empirical Higher Order Influence Function Estimators
• Root-n consistent semiparametric learning with high-dimensional nuisance functions under minimal sparsity
• On the Asymptotic Inadmissibility of Double Machine Learning Estimators Under Structure-Agnostic Models
• Quantum thermodynamics and semi-definite optimization
• Cross-Fitting and Fast Remainder Rates for Semiparametric Estimation
• Technical Report: Higher Order Influence Functions and Minimax Estimation of Nonlinear Functionals
• Higher Order Estimating Equations for High-dimensional Models
• Higher Order Targeted Maximum Likelihood Estimation
• Fisher Information in Kinetic Theory

Related papers

• Conformal Prediction for Dyadic Regression Under Complex Missingness
• Sharp Asymptotic Theory of Maximum Likelihood Estimation for Gaussian Processes with an RBF Kernel
• Bentkus-type asymptotic e-values
• High-Dimensional Asymptotics of Differentially Private PCA
• A proximal gradient algorithm for composite log-concave sampling
• KL Convergence Guarantees for Score diffusion models under minimal data assumptions