Adaptive Nonparametric Statistical Theory and Implementation

Abstract

In the world of Big Data, statistical tools are used to capture information that form the theoretical foundation for predictions, policies and decision making. Across the board, researchers rely on the accuracy of these statistical tools in translating swarms of data into meaningful insights. It is important, then, in practice, to have a clear understanding of the contextual applications, reliability and limitation of any statistical analysis tool that is used by researchers in inference and decision making. The goal of this dissertation is to influence data analysis procedures and give researchers a better understanding of how well different statistical tools work in various practical settings. This work is largely informed by how estimation tools are implemented as well as the types of data that are available for analysis. The first chapter uses local polynomial methods to establish a one-step boundary- adaptive conditional density estimator that is easy to implement. Pointiwse and uniform properties of the estimator are studied, providing concentration of the estimator, both, in probability and distribution. The results of this chapter also provide a general framework under which functional of the estimator can also be approximated. Particularly, the construction of uniform confidence bands is studied in this chapter. To aid the use of the minimax-optimal conditional density estimator, the second chapter of this thesis discusses the implementation of the estimator in a software package, lpcde, built for R. This chapter delves into the computational considerations made for efficient density estimation and confidence band construction. The chapter provides tools for how other researchers may employ the open-source package to their own applications through a simulated study illustration. The final chapter of this thesis looks at another class of popular adaptive estimators, decision trees. Specifically, this chapter provides the first theoretical concentration results for a general class of decision tree and forest estimators in the regression setting. This chapter also showcases the settings in which tree-based estimators can achieve the same rates of convergence as neural networks, providing a powerful argument for the continued use of decision trees in empirical analyses.