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Title: Statistical inference of robust regression with contaminated errors Authors:  Zhao Ren - University of Pittsburgh (United States) [presenting]
Wenxin Zhou - University of California San Diego (United States)
Peiliang Zhang - University of Pittsburgh (United States)
Abstract: The robust estimation and inference problems for linear regression are studied in the increasing dimension regime. Given a random design, we consider the conditional distributions of error terms are contaminated by some arbitrary distribution (possibly depending on the covariates) with proportion $\epsilon$ but otherwise can also be heavy-tailed and asymmetric. We show that simple robust $M$-estimators such as Huber and smoothed Huber, with an additional intercept added in the model, can achieve the minimax rates of convergence under the $l_2$ loss. In addition, two types of confidence intervals with root-$n$ consistency are provided by a multiplier bootstrap technique when the necessary condition on contamination proportion $\epsilon=o(1/\sqrt{n})$ holds. For a larger $\epsilon$, we further propose a debiasing procedure to reduce the potential bias caused by contamination, and prove the validity of the debiased confidence interval. At last, we extend our methods to the communication-efficient distributed estimation and inference setting. A comprehensive simulation study exhibits the effectiveness of our proposed inference procedures.