Adaptive penalized quantile regression for high dimensional data

We propose a new adaptive L1 penalized quantile regression estimator for high-dimensional sparse regression models with heterogeneous error sequences. We show that under weaker conditions compared with alternative procedures, the adaptive L1 quantile regression selects the true underlying model with probability converging to one, and the unique estimates of nonzero coefficients it provides have the same asymptotic normal distribution as the quantile estimator which uses only the covariates with non-zero impact on the response. Thus, the adaptive L1 quantile regression enjoys oracle properties. We propose a completely data driven choice of the penalty level λnλn, which ensures good performance of the adaptive L1 quantile regression. Extensive Monte Carlo simulation studies have been conducted to demonstrate the finite sample performance of the proposed method.

► We propose a quantile regression with a fully adaptive L1 penalty function.
► We show that the proposed estimator possesses oracle properties.
► We propose a new data-driven procedure to select the penalty level to achieve the oracle rate.
► Numerical simulations demonstrate the satisfactory performance of the proposed estimator.