Confidence Intervals for Linear Models with Arbitrary Noise Contamination

TL;DR

This paper introduces a new method for constructing confidence intervals for linear regression coefficients under arbitrary noise contamination, achieving optimal length without knowing the contamination level.

Contribution

It develops a novel Z-estimation based algorithm that provides valid, adaptive confidence intervals in contaminated linear models without prior contamination knowledge.

Findings

Confidence intervals have valid coverage under all contamination distributions.

Achieves an optimal length of order O(1/√(n(1-ε)^2)), matching known contamination scenarios.

Contrasts with Gaussian models by avoiding adaptation costs in robust interval estimation.

Abstract

We study confidence interval construction for linear regression under Huber's contamination model, where an unknown fraction of noise variables is arbitrarily corrupted. While robust point estimation in this setting is well understood, statistical inference remains challenging, especially because the contamination proportion is not identifiable from the data. We develop a new algorithm that constructs confidence intervals for individual regression coefficients without any prior knowledge of the contamination level. Our method is based on a Z-estimation framework using a smooth estimating function. The method directly quantifies the uncertainty of the estimating equation after a preprocessing step that decorrelates covariates associated with the nuisance parameters. We show that the resulting confidence interval has valid coverage uniformly over all contamination distributions and attains an optimal length of order $O(1/\sqrt{n(1-\epsilon)^2})$, matching the rate achievable when the contamination proportion $\epsilon$ is known. This result stands in sharp contrast to the adaptation cost of robust interval estimation observed in the simpler Gaussian location model.