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

近年来，Tran密度和依赖关系下的回归问题越来越受到人们的关注。本研究涵盖两个广泛但相关的领域：非参数密度估计和非参数回归。所处理的依赖关系是序列型或空间型的。序列相关随机变量的非参数泛函估计的最优性结果大多为渐近型。在小样本情况下如何做的实际问题是一个非常值得关注的问题，并且仍然提出了基本的挑战。本研究的目的是将现有的理论延伸到实践者的需求。目标是发展非参数密度和回归估计的有限样本理论。本建议的另一个主要特点是将时间序列的非参数密度和回归的某些结果扩展到随机场。不同的密度估计和固定和随机回归设计进行了研究。与传统的统计理论相比，非参数估计提供了一种灵活的数据分析方法，而不需要对所调查的过程有详细的了解或假设。研究的目的是发展非参数统计理论，以更好地服务于从业者。调查的适用性将通过具体和现实生活中的例子来证明。长期目标是发展非参数技术用于数字图像处理，甚至用于分析空间和时间变化的数据，例如从不同时间拍摄的一系列数字化照片中获得的数据。该研究也有可能应用于分析在不同空间和时间点不规则收集的数据。由此产生的方法可能对处理环境数据以及地质学、土壤科学和气象学数据很重要。＊＊＊

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. ***