Scalable and Efficient Statistical Inference with Estimating Functions in the MapReduce Paradigm for Big Data

The theory of statistical inference along with the strategy of divide-and-conquer for large- scale data analysis has recently attracted considerable interest due to great popularity of the MapReduce programming paradigm in the Apache Hadoop software framework. The central analytic task in the development of statistical inference in the MapReduce paradigm pertains to the method of combining results yielded from separately mapped data batches. One seminal solution based on the confidence distribution has recently been established in the setting of maximum likelihood estimation in the literature. This paper concerns a more general inferential methodology based on estimating functions, termed as the Rao-type confidence distribution, of which the maximum likelihood is a special case. This generalization provides a unified framework of statistical inference that allows regression analyses of massive data sets of important types in a parallel and scalable fashion via a distributed file system, including longitudinal data analysis, survival data analysis, and quantile regression, which cannot be handled using the maximum likelihood method. This paper investigates four important properties of the proposed method: computational scalability, statistical optimality, methodological generality, and operational robustness. In particular, the proposed method is shown to be closely connected to Hansen's generalized method of moments (GMM) and Crowder's optimality. An interesting theoretical finding is that the asymptotic efficiency of the proposed Rao-type confidence distribution estimator is always greater or equal to the estimator obtained by processing the full data once. All these properties of the proposed method are illustrated via numerical examples in both simulation studies and real-world data analyses.