Asymptotic normality of robust M-estimators with convex penalty

Research output: Contribution to journalArticlepeer-review


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, p/n ≤ γ for some fixed constant γ > 0. 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 (p/n → 0) or high (p/n ≤ γ) 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.

Original languageEnglish (US)
Pages (from-to)5591-5622
Number of pages32
JournalElectronic Journal of Statistics
Issue number2
StatePublished - 2022

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • M-estimator
  • Robust estimation
  • Stein’s formula
  • asymptotic normality
  • bias-correction
  • confidence Intervals
  • high-dimensional statistics


Dive into the research topics of 'Asymptotic normality of robust M-estimators with convex penalty'. Together they form a unique fingerprint.

Cite this