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, 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 language | English (US) |
|---|---|
| Pages (from-to) | 5591-5622 |
| Number of pages | 32 |
| Journal | Electronic Journal of Statistics |
| Volume | 16 |
| Issue number | 2 |
| DOIs | |
| State | Published - 2022 |
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- M-estimator
- Robust estimation
- Stein’s formula
- asymptotic normality
- bias-correction
- confidence Intervals
- high-dimensional statistics