Mathematical Sciences: Nonparametric Density & Regression Estimation for Dependent Random Variables

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Tran Density and regression under dependence have recently received increasing attention. This research covers two broad but related areas, nonparametric density estimation and nonparametric regression. The dependence dealt with is of serial or spatial type. Optimality results on nonparametric functional estimation for serially dependent random variables have been mostly of the asymptotic type. The practical question as to what to do in the small sample case is of great concern and still poses basic challenges. The purpose of this research is to extend existing theory toward the needs of practitioners. The objective is the development of a finite sample theory for nonparametric density and regression estimators. Another major feature of the current proposal is the extension of certain results on nonparametric density and regression known for time series to random fields. Different density estimators and both fixed and random regression designs are to be investigated. In contrast to traditional statistical theory, nonparametric estimation gives a flexible approach to data analysis without requiring detailed knowledge or assumptions about the process under investigation. The goal of the research is to develop nonparametric statistical theory to better serve practitioners. The applicability of the investigation will be demonstrated by concrete and real-life examples. The long-range goal is to develop nonparametric techniques for digital image processing, and even for the analysis of data which are both space and time varying, such as data obtained from a series of digitized photographs taken at different times. The proposed research also has potential application to the analysis of data collected irregularly at different space and time points. The resulting methodology may prove important for processing environmental data and data from geology, soil science and meteorology. ***