Asymptotic normality of robust M-estimators with convex penalty

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 same order as sample size n, i.e, p∕n≤γ some fixed constant γ>0. The requires a bias correction and holds M-estimator large class loss functions including Huber its smoothed versions regularized strongly penalty. variance that characterizes width resulting confidence intervals estimated data-driven quantities. estimate adapts automatically to low (p∕n→0) or high (p∕n≤γ) dimensions does not involve proximal operators seen previous works on M-estimators. For loss, has simple expression involving an effective degrees-of-freedom well size. case Elastic-Net studied details simulation study confirms theoretical findings. follow from Stein formulae high-dimensional random vectors sphere developed which are independent interest.