Robust communication-efficient distributed composite quantile regression and variable selection for massive data

Statistical analysis of massive data is becoming more and more common. Distributed composite quantile regression (CQR) for massive data is proposed in this paper. Specifically, the global CQR loss function is approximated by a surrogate one on the first machine, which relates to the local data only through their gradients, then the estimator is obtained on the first machine by minimizing the surrogate loss. Because the gradients of local datasets can be efficiently communicated, the communication cost is significantly reduced. In order to reduce the computational burdens, the induced smoothing method is applied. Theoretically, the resulting estimator is proved to be statistically as efficient as the global CQR estimator. What is more, as a direct application, a smooth-threshold distributed CQR estimating equations for variable selection is proposed. The new methods inherit the robustness and efficiency advantages of CQR. The promising performances of the new methods are supported by extensive numerical examples and real data analysis. (C) 2021 Elsevier B.V. All rights reserved.