Sparse and debiased lasso estimation and inference for high-dimensional composite quantile regression with distributed data

Abstract

We consider the data are inherently distributed and focus on statistical learning in the presence of heavy-tailed and/or asymmetric errors. The composite quantile regression (CQR) estimator is a robust and efficient alternative to the ordinary least squares and single quantile regression estimators. Based on the aggregated and communication-efficient approaches, we propose two classes of sparse and debiased lasso CQR estimation and inference methods. Specifically, an aggregated $\ell _1$-penalized CQR estimator and a $\ell _1$-penalized communication-efficient CQR estimator are obtained firstly. To construct confidence intervals and make hypothesis testing, a unified debiasing framework based on smoothed decorrelated score equations is introduced to eliminate biases caused by lasso penalty. Finally, a hard-thresholding method is employed to ensure that the debiased lasso estimators are sparse. The convergence rates and asymptotic properties of the proposed estimators are established and their performance is evaluated through simulations and a real-world dataset.