Modern Sieve Estimators for Nonparametric Problems: Streaming Data and High-dimensional Data

Abstract

Estimation of a regression function, linking a set of features to an outcome of interest, is a fundamental statistical task. This dissertation focuses on the application of sieve estimators in modern statistical learning problems. The method of sieves, or estimation via basis expansion, has its roots in Fourier analysis. In the past decades, it has achieved much success in smaller sample size, lower dimensional data science problems. In this dissertation, we will demonstrate its effectiveness in modern statistical learning settings. Sieve estimators can achieve statistical and computational optimality (almost) simultaneously, which makes them very suitable for online and/or large scale nonparametric estimation tasks. Sieve estimators can also be applied to high-dimensional nonparametric problems. They can effectively alleviate the “curse of dimensionality” by leveraging additional structures such as feature sparsity. For each topic covered in this dissertation, we will present both theoretical discussion and a variety of numerical examples.