In recent years, many methodologies for distributed data have been developed. However, there are two problems. First, most of these methods require the data to be randomly and uniformly distributed across different machines. Second, the methods are mainly not robust. To solve these problems, we propose a distributed pilot modal regression estimator, which achieves robustness and can adapt when the data are stored nonrandomly. First, we collect a random pilot sample from different machines; then, we approximate the global MR objective function by a communication-efficient surrogate that can be efficiently evaluated by the pilot sample and the local gradients. The final estimator is obtained by minimizing the surrogate function in the master machine, while the other machines only need to calculate their gradients. Theoretical results show the new estimator is asymptotically efficient as the global MR estimator. Simulation studies illustrate the utility of the proposed approach.

近年来，已经开发了许多用于分布式数据的方法。但是，有两个问题。首先，这些方法中的大多数都要求数据随机且均匀地分布在不同的机器上。其次，这些方法主要是不鲁棒的。为了解决这些问题，我们提出了一种分布式导频模态回归估计器，它实现了鲁棒性，并且可以在数据非随机存储时适应。首先，我们从不同的机器上随机收集试点样本；然后，我们通过一个通信高效的代理来近似全局 MR 目标函数，该代理可以通过引导样本和局部梯度进行有效评估。最终的估计量是通过最小化主机中的代理函数得到的，而其他机器只需要计算它们的梯度。理论结果表明，新估计器作为全局 MR 估计器是渐近有效的。仿真研究说明了所提出方法的实用性。