Scale calibration for high-dimensional robust regression

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