Asymptotic normality of robust $M$-estimators with convex penalty

Understand

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, $p/n\le\gamma$ for some fixed constant $\gamma>0$.

• 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 ($p/n\to0)$ or high ($p/n \le \gamma$) dimensions and does not involve the proximal operators seen in previous works on asymptotic normality of M-estimators.