Smoothing and nonparametric tests

数据分析的一个基本问题是检查统计模型与观测数据的拟合程度。非参数平滑和测试方法为模型的建立和检验以及参数模型的拟合优度检验提供了方便而有力的工具。平滑是现代非参数统计中的一个重要课题。密度估计是非参数平滑、检验和许多重要统计问题中的一个基本问题。一种好的密度估计方法对于开发基于似然比的平滑和强大的非参数检验的新方法至关重要。密度估计一直是统计文献中的一个热点问题。最常用的方法是核密度估计，其关键问题是选择带宽以平衡平滑性和准确性。对于非参数模型的一般核平滑方法，带宽选择也是至关重要的。或大或小的带宽将使内核模型平滑或不平滑。快速简单的方法易于应用，但对于复杂的模型效果较差。另一方面，现有的高技术方法存在计算难度大，有时没有解的问题。数据驱动的带宽要求高，应用简单，适合各种模型。一种可能的解决方案是使用初始固定（全局）带宽来构建可变（局部）带宽，或者使用其他局部平滑技术。研究结果对非参数建模和测试新技术的发展具有重要意义。

A fundamental problem in data analysis is to check how well a statistical model fits observed data. Nonparametric smoothing and testing methods provide convenient and powerful tools for model building and checking, as well as goodness-of-fit tests for parametric models. Smoothing is a major subject in modern nonparametric statistics. Density estimation is a basic issue in nonparametric smoothing, testing, and many important statistical problems. A good method of density estimation is essential for developing new methodologies of smoothing and powerful nonparametric tests based on the likelihood ratio. Density estimation has been a hot topic in the statistical literature. The most popular method is kernel density estimation, for which the critical problem is bandwidth selection for balancing smoothness and accuracy. Bandwidth selection is also crucial to general kernel smoothing methods for nonparametric models. A large or small bandwidth will over or under smooth the kernel model. Quick-simple methods are easy to apply but perform poor for complicated models. On the other hand, the existing hi-tech methods have computational difficulty, and do not have solutions occasionally. A data-driven bandwidth is highly demanded, which is easy to apply and suits for various kinds of models. One possible solution is to use an initial fixed (global) bandwidth to build a variable (local) bandwidth, or to use other localized smoothing techniques. The results of this research are useful in developing new technologies for nonparametric modeling and testing.