Large-Scale Global and Simultaneous Inference: Estimation and Testing in Very High Dimensions

Due to rapid technological advances, researchers are now able to collect and analyze ever larger data sets. Statistical inference for big data often requires solving thousands or even millions of parallel inference problems simultaneously. This poses significant challenges and calls for new principles, theories, and methodologies. This review provides a selective survey of some recently developed methods and results for large-scale statistical inference, including detection, estimation, and multiple testing. We begin with the global testing problem, where the goal is to detect the existence of sparse signals in a data set, and then move to the problem of estimating the proportion of nonnull effects. Finally, we focus on multiple testing with false discovery rate (FDR) control. The FDR provides a powerful and practical approach to large-scale multiple testing and has been successfully used in a wide range of applications. We discuss several effective data-driven procedures and also present efficient strategies to handle various grouping, hierarchical, and dependency structures in the data.