Debiased inference for a covariate-adjusted regression function

TL;DR

This paper introduces a debiased local linear estimator for covariate-adjusted regression functions, enabling valid inference and confidence bands in causal dose-response analysis without undersmoothing.

Contribution

It proposes a novel debiased estimator that achieves optimal convergence rates and allows flexible, data-adaptive nuisance function estimation for nonparametric causal inference.

Findings

Estimator converges to a normal distribution for pointwise inference.

Constructs valid confidence intervals and uniform confidence bands.

Demonstrates practical effectiveness through simulations and air pollution study.

Abstract

In this article, we study nonparametric inference for a covariate-adjusted regression function. This parameter captures the average association between a continuous exposure and an outcome after adjusting for other covariates. In particular, under certain causal conditions, this parameter corresponds to the average outcome had all units been assigned to a specific exposure level, known as the causal dose-response curve. We propose a debiased local linear estimator of the covariate-adjusted regression function, and demonstrate that our estimator converges pointwise to a mean-zero normal limit distribution. We use this result to construct asymptotically valid confidence intervals for function values and differences thereof. In addition, we use approximation results for the distribution of the supremum of an empirical process to construct asymptotically valid uniform confidence bands. Our methods do not require undersmoothing, permit the use of data-adaptive estimators of nuisance functions, and our estimator attains the optimal rate of convergence for a twice differentiable function. We illustrate the practical performance of our estimator using numerical studies and an analysis of the effect of air pollution exposure on cardiovascular mortality.