Progress in nonparametric minimax estimation and high dimensional hypothesis testing

Abstract

This dissertation is divided into two parts. In the first part, we study minimax estimation of functions and functionals in nonparametric regression models. The investigation of statistical limits in such models deepens theoretical understanding in related problems and leads to new probabilistic tools and methodologies of broader interest. In the second part, we study the asymptotics in some high dimensional testing problems involving the Gaussian distribution, such as the Gaussian sequence model with convex constraint and testing of covariance matrices. A general framework is developed to analyze the power behavior of test statistics via accurate non-asymptotic expansions.