Two-stage communication-efficient distributed sparse M-estimation with missing data

Distributed estimation based on different sources of observations has drawn attention in the modern statistical learning. When the distributed data are missing at random, we propose a two-stage -penalized communication-efficient surrogate likelihood (CSL) algorithm based on inverse probability weighting to eliminate the estimation bias caused by the missing data and construct sparse distributed M-estimator simultaneously. In the first stage, we consider a parametric propensity model and directly apply the -penalized CSL method to obtain an efficient and sparse distributed estimator of the propensity parameter. In the second stage, we construct an IPW-based -penalized CSL loss function to eliminate the bias and obtain the sparse M-estimation. The finite-sample performance of the estimators is studied through simulation, and an application to house sale prices data set is also presented.