Communication‐efficient low‐dimensional parameter estimation and inference for high‐dimensional Lp$$ {L}^p $$‐quantile regression

Abstract

AbstractThe ‐quantile regression generalizes both quantile regression and expectile regression, and has become popular for its robustness and effectiveness especially when . In this paper, we consider the data that are inherently distributed and propose two distributed ‐quantile regression estimators for a preconceived low‐dimensional parameter in the presence of high‐dimensional extraneous covariates. To handle the impact of high‐dimensional nuisance parameters, we first investigate regularized projection score for estimating low‐dimensional parameter of main interest in ‐quantile regression. To deal with the distributed data, we further propose two communication‐efficient surrogate projection score estimators and establish their theoretical properties. The finite‐sample performance of the proposed estimators is studied through simulations and an application to Communities and Crime data set is also presented.