Relative error-based distributed estimation in growing dimensions

This paper studies the estimation problem of multiplicative models for large-scale positive response data with growing dimensions. First, we propose a communication-efficient least product relative error estimator which is the minimizer of a surrogate loss function that approximates the global least product relative error loss. Then, a practically efficient distributed Newton-Raphson algorithm is proposed to solve the problem. Theoretically, we show that the distributed estimator achieves the same statistical efficiency as the global estimator under regularity conditions when the dimension is fixed and increases with the local sample size, respectively. Second, by incorporating an adaptive lasso penalty into the surrogate loss, we develop a communication- efficient penalized least product relative error estimator for high-dimensional variable selection on massive positive response data. Accordingly, we propose a distributed algorithm based on the alternating direction method of multipliers. It is shown that the distributed penalized estimator has the oracle property in the growing-dimensional setting. Finally, extensive simulations and two real-world applications are conducted to demonstrate the superiority of our proposal.