Some computationally efficient methods in statistics and their applications in parameter estimation and hypotheses testing

Abstract

Parameter estimation and hypotheses testing are two fundamental problems in statistics. Many existing methods have been developed for the problems with moderate amount of data. Unfortunately, some of those methods could be computationally costly or even infeasible when the volume of data is high. This dissertation is an attempt to fulfill the needs for computationally efficient methods in statistics. The first part of this dissertation describes a one-step approach to enhance an existing simple averaging estimator for distributed statistical inference, which achieve the same convergence rate with the estimator using centralized data. In the second part, we develop an efficient algorithm with reduced computational complexity for distance covariance and apply this new algorithm to derive a test of independence, which enjoys nearly the same asymptotic efficiency with the state-of-the-art distance covariance. The third part is a statistically and computationally efficient two-sample test based on energy statistics and random projections.