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

TL;DR

This paper develops a robust hypothesis testing method for penalized estimators in complex models, using cross-fitting to calibrate covariance estimation and ensure valid inference.

Contribution

It introduces a novel covariance calibration technique via cross-fitting for hypothesis testing with penalized estimating equations.

Findings

The test statistic converges to a chi-squared distribution.

Cross-fitting improves robustness against covariance misspecification.

The method maintains $\

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 $\chi^2$ 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.