Modal regression is a good alternative of the mean regression and likelihood based methods, because of its robustness and high efficiency. A robust communication-efficient distributed modal regression for the distributed massive data is proposed in this paper. Specifically, the global modal regression objective function is approximated by a surrogate one at the first machine, which relates to the local datasets only through gradients. Then the resulting estimator can be obtained at the first machine and other machines only need to calculate the gradients, which can significantly reduce the communication cost. Under mild conditions, the asymptotical properties are established, which show that the proposed estimator is statistically as efficient as the global modal regression estimator. What is more, as a specific application, a penalized robust communication-efficient distributed modal regression variable selection procedure is developed. Simulation results and real data analysis are also included to validate our method.

模态回归由于其鲁棒性和高效性，是均值回归和基于似然法的一种很好的选择。针对分布式海量数据，提出了一种鲁棒的，通信效率高的分布式模态回归方法。具体而言，全局模态回归目标函数在第一台机器上由代理人近似，该代理人仅通过梯度与局部数据集相关。然后，可以在第一台机器上获得最终的估计器，而其他机器仅需要计算梯度，就可以大大降低通信成本。在温和条件下，建立了渐近性质，这表明所提出的估计量在统计上与全局模态回归估计量一样有效。此外，作为特定的应用程序，开发了一种惩罚性的，鲁棒的，通信效率高的分布式模态回归变量选择程序。仿真结果和真实数据分析也包括在内，以验证我们的方法。