Abstract
This paper develops asymptotic normality results for individual coordinates of robust M-estimators with convex penalty in high-dimensions, where the dimension p is at most of the same order as the sample size n, i.e, for some fixed constant . The asymptotic normality requires a bias correction and holds for most coordinates of the M-estimator for a large class of loss functions including the Huber loss and its smoothed versions regularized with a strongly convex penalty.
The asymptotic variance that characterizes the width of the resulting confidence intervals is estimated with data-driven quantities. This estimate of the variance adapts automatically to low ( or high () dimensions and does not involve the proximal operators seen in previous works on asymptotic normality of M-estimators. For the Huber loss, the estimated variance has a simple expression involving an effective degrees-of-freedom as well as an effective sample size. The case of the Huber loss with Elastic-Net penalty is studied in details and a simulation study confirms the theoretical findings. The asymptotic normality results follow from Stein formulae for high-dimensional random vectors on the sphere developed in the paper which are of independent interest.
Funding Statement
P.C. Bellec was partially supported by the NSF Grants DMS-1811976 and DMS-1945428. C.-H. Zhang was partially supported by the NSF Grants DMS-1721495, IIS-1741390, CCF-1934924, DMS-2052949 and DMS-2210850.
Citation
Pierre C. Bellec. Yiwei Shen. Cun-Hui Zhang. "Asymptotic normality of robust M-estimators with convex penalty." Electron. J. Statist. 16 (2) 5591 - 5622, 2022. https://doi.org/10.1214/22-EJS2065
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