Quantile regression feature selection and estimation with grouped variables using Huber approximation

This paper considers model selection and estimation for quantile regression with a known group structure in the predictors. For the median case the model is estimated by minimizing a penalized objective function with Huber loss and the group lasso penalty. While, for other quantiles an M-quantile approach, an asymmetric version of Huber loss, is used which approximates the standard quantile loss function. This approximation allows for efficient implementation of algorithms which rely on a differentiable loss function. Rates of convergence are provided which demonstrate the potential advantages of using the group penalty and that bias from the Huber-type approximation vanishes asymptotically. An efficient algorithm is discussed, which provides fast and accurate estimation for quantile regression models. Simulation and empirical results are provided to demonstrate the effectiveness of the proposed algorithm and support the theoretical results.