Hypothesis Testing for Penalized Estimating Equations with Cross-Fitted Covariance Calibration

2026-04-06

AI Summary

This study addresses the challenge of conducting inference in settings where the marginal distributions of multivariate responses are difficult to specify accurately—such as in longitudinal data or high-dimensional heteroscedastic regression—under the assumption that the conditional mean model is correctly specified. The authors develop a √n-consistent estimator via a penalized estimating equation and propose a hypothesis testing procedure focused on low-dimensional subvectors of parameters. A key innovation is the introduction of a cross-fitted covariance calibration mechanism, which substantially reduces the sensitivity of the test to misspecification of the nuisance covariance function. The resulting test statistic follows an asymptotic chi-squared distribution, achieving robustness by controlling Type I error while significantly enhancing statistical power for efficient and reliable inference.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We study hypothesis testing for penalized estimators in settings where the full marginal distribution of a multivariate response is difficult to specify, such as longitudinal data with correlated measurements or high-dimensional heteroscedastic regression. Assuming that the conditional mean model is correctly specified, we establish that the penalized estimating equations admit a $\sqrt{n}$-consistent solution, even when the working covariance structure is misspecified. Our inferential target is a low-dimensional subvector of parameters associated with the mean model. We show that the resulting test statistic converges to a $χ^2$ distribution, and that its asymptotic power depends on the nuisance covariance function. To mitigate this dependence, we propose estimating the covariance function via cross-fitting, which provides a calibrated and robust procedure for inference.

Problem

• hypothesis testing
• penalized estimating equations
• covariance misspecification
• high-dimensional regression
• longitudinal data

Innovation

• penalized estimating equations
• cross-fitting
• covariance calibration
• hypothesis testing
• robust inference