Scattered Data and Aggregated Inference

Scattered Data and Aggregated Inference (SDAI) represents a class of problems where data cannot be at a centralized location, while modeling and inference is pursued. Distributed statistical inference is a technique to tackle a type of the above problem, and has recently attracted enormous attention. Many existing work focus on the averaging estimator, e.g., Zhang et al. (2013) and many others. In this chapter, we propose a one-step approach to enhance a simple-averaging based distributed estimator. We derive the corresponding asymptotic properties of the newly proposed estimator. We find that the proposed one-step estimator enjoys the same asymptotic properties as the centralized estimator. The proposed one-step approach merely requires one additional round of communication in relative to the averaging estimator; so the extra communication burden is insignificant. In finite-sample cases, numerical examples show that the proposed estimator outperforms the simple averaging estimator with a large margin in terms of the mean squared errors. A potential application of the one-step approach is that one can use multiple machines to speed up large-scale statistical inference with little compromise in the quality of estimators. The proposed method becomes more valuable when data can only be available at distributed machines with limited communication bandwidth. We discuss other types of SDAI problems at the end.