Confidence Intervals for Linear Models with Arbitrary Noise Contamination

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.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-11-10

• Fields of study

  Mathematics

• Identifiers
  arXiv 2511.07605
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Sharp Concentration of Simple Random Tensors

  2025influential reference
• Adaptive Robust Confidence Intervals

  2024influential reference
• Residual permutation test for regression coefficient testing

  2022influential reference
• Consistent Estimation for PCA and Sparse Regression with Oblivious Outliers

  2021influential reference
• Robust Universal Inference

  2021cited by this paper
• The HulC: Confidence Regions from Convex Hulls

  2021cited by this paper
• Model Repair: Robust Recovery of Over-Parameterized Statistical Models

  2020influential reference
• Consistent regression when oblivious outliers overwhelm

  2020influential reference
• Adaptive Hard Thresholding for Near-optimal Consistent Robust Regression

  2019influential reference
• An assumption-free exact test for fixed-design linear models with exchangeable errors

  2019influential reference
• Consistent Robust Regression

  2017influential reference
• Adaptive estimation of high-dimensional signal-to-noise ratios

  2016cited by this paper
• Asymptotics for high dimensional regression M-estimates: fixed design results

  2016cited by this paper
• Linear Hypothesis Testing in Dense High-Dimensional Linear Models

  2016cited by this paper
• Robust covariance and scatter matrix estimation under Huber’s contamination model

  2015cited by this paper
• Robust Regression via Hard Thresholding

  2015influential reference
• Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity

  2015cited by this paper
• Convergence of the Huber Regression M-Estimate in the Presence of Dense Outliers

  2014influential reference
• Asymptotic behavior of unregularized and ridge-regularized high-dimensional robust regression estimators : rigorous results

  2013cited by this paper
• Confidence intervals and hypothesis testing for high-dimensional regression

  2013cited by this paper
• On robust regression with high-dimensional predictors

  2013cited by this paper
• On asymptotically optimal confidence regions and tests for high-dimensional models

  2013cited by this paper
• High dimensional robust M-estimation: asymptotic variance via approximate message passing

  2013cited by this paper
• Asymptotic normality and optimalities in estimation of large Gaussian graphical models

  2013cited by this paper
• Optimal M-estimation in high-dimensional regression

  2013cited by this paper
• Confidence intervals for low dimensional parameters in high dimensional linear models

  2011cited by this paper
• Introduction to the non-asymptotic analysis of random matrices

  2010influential reference
• BOOTSTRAPPING REGRESSION MODELS WITH MANY PARAMETERS

  2008cited by this paper
• Bootstrapping Regression Models

  2002cited by this paper
• Probability Inequalities for Sums of Bounded Random Variables

  1994cited by this paper
• Bootstrap and Wild Bootstrap for High Dimensional Linear Models

  1993cited by this paper
• Asymptotics with increasing dimension for robust regression with applications to the bootstrap

  1989cited by this paper
• Asymptotic behavior of M-estimators of p regression parameters when p

  1985cited by this paper
• Asymptotic Behavior of $M$-Estimators of $p$ Regression Parameters when $p^2/n$ is Large. I. Consistency

  1984cited by this paper
• On robust estimation of the location parameter

  1980cited by this paper
• ASYMPTOTIC BEHAVIOR OF M-ESTIMATORS FOR THE LINEAR MODEL

  1979cited by this paper
• Robust Regression: Asymptotics, Conjectures and Monte Carlo

  1973cited by this paper
• Non-Parametric Estimation. I. Validation of Order Statistics

  1945cited by this paper

• No citing papers are available for this paper.