Open Access
2021 Scale calibration for high-dimensional robust regression
Po-Ling Loh
Author Affiliations +
Electron. J. Statist. 15(2): 5933-5994 (2021). DOI: 10.1214/21-EJS1936


We present a new method for robust high-dimensional linear regression when the scale parameter of the additive errors is unknown. The proposed estimator is based on a penalized Huber M-estimator, for which theoretical bounds on the estimation error have recently been proposed in high-dimensional statistics literature. However, the variance of the error term in the linear model is intricately connected to the optimal parameter used to define the shape of the Huber loss. Our main idea is to use an adaptive technique, based on Lepski’s method, to overcome the difficulties of solving a joint nonconvex optimization problem with respect to the location and scale parameters. Furthermore, by including a weight term in the definition of the M-estimator, our consistency results hold even when the covariates are heavy-tailed. We then derive asymptotic normality of a one-step estimator constructed from the penalized Huber estimator, which can be used to construct confidence regions for subsets of coordinates. The one-step estimator is shown to be semiparametrically efficient when the covariates are sub-exponential. Our results substantially generalize previous work on high-dimensional inference, derived under sub-Gaussian assumptions on both the covariate and error distributions.

Funding Statement

Part of this work was completed while the author was visiting the Isaac Newton Institute in Cambridge, UK. The author was supported by NSF grant DMS-1749857.


The author would like to thank Ezequiel Smucler for sharing the archaeological dataset used in the simulations. The author also thanks the AE and anonymous reviewers for thoughtful feedback which greatly improved the manuscript.


Download Citation

Po-Ling Loh. "Scale calibration for high-dimensional robust regression." Electron. J. Statist. 15 (2) 5933 - 5994, 2021.


Received: 1 April 2020; Published: 2021
First available in Project Euclid: 27 December 2021

Digital Object Identifier: 10.1214/21-EJS1936

Primary: 62F10
Secondary: 62J07

Keywords: Adaptive scale estimation , high-dimensional inference , Lepski’s method , one-step estimation , robust linear regression , Semiparametric efficiency

Vol.15 • No. 2 • 2021
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