Confidence Intervals for Linear Models with Arbitrary Noise Contamination

2025-11-10

AI Summary

This paper addresses the problem of constructing confidence intervals for linear regression coefficients under the Huber contamination model, where an unknown fraction ε of arbitrary outliers invalidates conventional inference. To tackle this challenge, we propose a novel Z-estimation-based algorithm: first, decorrelating covariates to mitigate design matrix correlation; second, employing a smoothed estimating function to achieve adaptive robust estimation of the regression parameters without prior knowledge of ε. The resulting confidence intervals attain uniform coverage at level (1−α) over all ε-contaminated distributions and achieve optimal width O(1/√[n(1−ε)²]), substantially improving upon existing adaptive methods. Our key contribution lies in the first unified integration of the Z-estimation framework, smoothing techniques, and covariate decorrelation—enabling consistent, efficient, and robust statistical inference under unknown contamination.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Constructing confidence intervals for linear regression with arbitrary noise contamination
• Developing inference methods without prior knowledge of contamination proportion
• Achieving optimal interval length matching known contamination rate performance

Innovation

• Confidence intervals without contamination level knowledge
• Z-estimation with smooth function decorrelating covariates
• Uniform coverage and optimal length over contamination distributions