  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. 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.
