Kernel Estimators in Complex Data Analysis

Abstract

Kernel estimators, including kernel density estimators and kernel regression estimators, have drawn great research interests in terms of both theoretical studies and applications since invention, due to their easy interpretation and flexibility to model data with complicated density curves/conditional mean curves. Even during this information age, when the datasets confronting us have become larger and more complicated, which seems to disfavor the use of kernel estimators because of the so- called “curse of dimensionality” phenomenon with nonparametric statistical methods, statisticians continue to propose sophisticated methods based on kernel estimators to handle complex data analysis problems. In this thesis, we talk about two such newly developed methodologies related to applications of kernel estimators. In Chapter 2, we develop a bandwidth (matrix) selector for multivariate kernel density estimators of level sets and highest density regions. We consider a different loss function from the one used in classical bandwidth selection problem and derive an asymptotic approximation to the corresponding risk function. A multi-stage plug-in bandwidth selection procedure is proposed to estimate the unknown quantity in the risk function and solve the optimal bandwidth. In Chapter 3, we propose a nonparametric doubly robust test for a continuous treatment effect, which involves applying a local polynomial estimator based nonparametric test to the pseudo outcomes aiming for causal effect estimation. We will also see under this framework how classical nonparametric statistical methods could collaborate with modern machine learning models to drive effective and reliable learning from complex real data.