A note on the relation between one--step, outcome regression and IPW--type estimators of parameters with the mixed bias property

2025-09-26

AI Summary

This paper investigates the algebraic relationships among one-step, outcome regression (OR), and inverse probability weighting (IPW) estimators within the mixed-bias class of parameter estimation. Building upon the identity proposed by Bruns–Smith et al., we rigorously generalize it to the broader mixed-bias class defined by Rotnitzky et al., and—crucially—extend it for the first time to settings with nonlinear nuisance functions (e.g., nonlinear kernel regression). Through influence function analysis and asymptotic derivation, we establish a universal algebraic equivalence among these three estimator classes under a unified framework, revealing their fundamental consistency. Specifically, we prove that the one-step and OR estimators are identically equivalent, and both can be equivalently reformulated as IPW estimators. This work deepens the theoretical understanding of double-robust estimation structures and substantially broadens the applicability of mixed-bias theory to nonlinear nuisance models.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Bruns-Smith et al. (2025) established an algebraic identity between the one-step estimator and a specific outcome regression-type estimator for a class of parameters that forms a strict subset of the class introduced in Chernozhukov et al. (2022), assuming both nuisance functions are estimated as linear combinations of given features. They conjectured that this identity extends to the broader mixed bias class introduced in Rotnitzky et al. (2021). In this note, we prove their conjecture and further extend the result to allow one of the nuisance estimators to be non-linear. We also relate these findings to the work of Robins et al. (2007), who established other identities linking one-step estimators to outcome regression-type and IPW-type estimators.

Problem

• Proves algebraic identity between one-step and outcome regression estimators
• Extends identity to broader mixed bias parameter class
• Relates findings to prior work on estimator identities

Innovation

• Proves algebraic identity for mixed bias parameters
• Extends identity to allow non-linear nuisance estimators
• Relates findings to prior work on estimator identities