Scale calibration for high-dimensional robust regression

We present a new method for robust high-dimensional linear regression when the scale parameter of the additive errors is unknown. The proposed estimator is based on a penalized Huber M-estimator, for which theoretical bounds on the estimation error have recently been proposed in high-dimensional statistics literature. However, the variance of the error term in the linear model is intricately connected to the optimal parameter used to define the shape of the Huber loss. Our main idea is to use an adaptive technique, based on Lepski's method, to overcome the difficulties of solving a joint nonconvex optimization problem with respect to the location and scale parameters. Furthermore, by including a weight term in the definition of the M-estimator, our consistency results hold even when the covariates are heavy-tailed. We then derive asymptotic normality of a one-step estimator constructed from the penalized Huber estimator, which can be used to construct confidence regions for subsets of coordinates. The one-step estimator is shown to be semiparametrically efficient when the covariates are sub-exponential. Our results substantially generalize previous work on high-dimensional inference, derived under sub-Gaussian assumptions on both the covariate and error distributions.