Optimal subsampling for $$L_p$$ -quantile regression via decorrelated score

Abstract

To balance robustness of quantile regression and effectiveness of expectile regression, we consider \(L_p\)-quantile regression models with large-scale data and develop a unified optimal subsampling method to downsize the data volume and reduce computational burden. For low-dimensional \(L_p\)-quantile regression models, two optimal subsampling probabilities based on the A- and L-optimality criteria are firstly proposed. For the preconceived low-dimensional parameter in high-dimensional \(L_p\)-quantile regression models, a novel optimal subsampling decorrelated score function is proposed to mitigate the effect from nuisance parameter estimation and then two optimal decorrelated score subsampling probabilities are provided. The asymptotic properties of two optimal subsample estimators are established. The finite-sample performance of the proposed estimators is studied through simulations, and an application to Beijing Air Quality Dataset is also presented.