平稳相依空间数据下基于经验似然的非参数统计推断

Empirical likelihood, introduced by Owen(1988, 1990), is an efficient method of nonparametric statistical inference, which can be well employed to construnct confidence intervals(or regions) for some parameters and to improve statistical inference efficently. Predecessors' research focuses on applying the empirical likelihood method to independent data or time series data. This project will mainly consider spatial data, the contents include the following aspects. 1) First, this project will employ the empirical likelihood method to construct confidence intervals for the nonparametric regression function under stationary and strongly mixing spatial data, and extend it to stationary and near-epoch dependent spatial data. 2) Second, this project will consider the reweighted Nadaraya-Watson estimator for the nonparametric regression function under stationary and strongly mixing spatial data, and extend it to stationary and near-epoch dependent spatial data. 3) Third, this project will employ the empirical likelihood method to construct confidence intervals for the conditional quantile in the prsence and absence of auxiliary information, respectively, under stationary and strongly mixing spatial data. In addition, this project will employ the empirical likelihood method to improve the kernel estimator of the conditional quantile with auxiliary information under stationary and strongly mixing spatial data. 4) Fourth, this project will employ the empirical likelihood method to construct confidence intervals for the nonparametric regression function under stationary and strongly mixing functional spatial data.

经验似然方法是 Owen(1988,1990) 提出的一种有效的非参数统计推断方法, 它可以很好地用来构造参数的置信区间(或区域)及有效地改进统计推断。前人的研究多集中将经验似然方法应用到独立或时间序列数据，本项目将主要针对空间数据。本项目拟应用经验似然方法来构造平稳强混合相依空间数据下非参数回归函数的置信区间，并将结果延伸到平稳近邻相依空间数据上；拟研究平稳强混合相依空间数据下非参数回归函数的复加权Nadaraya-Watson估计，并将结果延伸到平稳近邻相依空间数据上；拟应用经验似然方法来分别构造平稳强混合相依空间数据下含有附加信息和不含有附加信息时条件分位数的置信区间，以及用经验似然方法来改进含有附加信息时条件分位数的核估计；拟应用经验似然方法来构造平稳强混合相依函数型空间数据下非参数回归函数的置信区间。

本项目用通常的经验似然方法来研究平稳强混合相依空间数据下非参数回归函数和条件分位数的相关统计推断，具体包括：.1)用经验似然方法来构造非参数回归函数的置信区间，在一定条件下证明了经验似然比统计量渐近服从自由度为1的标准卡方分布。数值模拟研究表明经验似然方法比正态逼近方法表现更好。.2)用经验似然方法来构造非参数回归函数的复加权NW估计，在一定条件下证明了估计的渐近正态性。数值模拟研究表明在经验均方误差方面，复加权NW估计与局部线性估计类似，表现比NW估计更好。.3)用经验似然方法来构造含有辅助信息下条件分位数的改进核估计，在一定条件下证明了估计的渐近正态性。从渐近正态性结果可以发现，改进核估计的渐近方差小于或等于通常核估计的渐近方差。数值模拟研究表明，改进核估计比通常核估计表现更好。