Spline Smoothing and Nonparametric Regression

提案ID：DMS-0203243PI：Eubank标题：样条平滑和非参数回归ABSTRACT研究了许多非参数回归类型的问题。这些问题是通过在其求解方法中使用样条平滑来联系在一起的。所研究的具体问题包括：1)在标准平滑参数一致性渐近在零假设下不成立的情况下，使用样条平滑器测试参数回归模型的不适合性，2)在变系数模型中使用样条平滑器进行估计，3)部分线性模型的方差估计和异方差检验，4)具有线性和非线性参数的非线性样条平滑问题的计算方法，5)自适应选择正则化参数用于不适定积分方程样条曲线光顺；6)等式约束局部多项式光滑器的计算和大样本性质及其在Copula密度估计中的应用。本研究项目研究的问题涉及回归分析，它代表了研究变量之间关系的标准统计方法。回归分析的经典方法假设变量集合之间的关系形式是已知的，除了必须从数据中估计的一些未知参数之外。该项目使用更现代的技术，使用灵活的或非参数的曲线拟合方法来产生估计器以及评估参数模型的有效性。新的估计方法被开发用于几种设置，包括时变系数模型和部分线性模型。时变系数模型提供了参数模型的一般化，其中回归关系中的参数被允许作为诸如时间的某个其他变量的函数而演变。这种类型的模型在许多情况下是有用的，例如用于分析来自纵向案例研究的数据，以及用于预测彩票销售作为中奖水平的函数。部分线性模型提供了参数和非参数方法的混合，其中一些变量的回归关系可以参数建模，而其他变量必须使用灵活的非参数技术来处理。后一种类型的模型被发现很有用，例如，在模拟农业田间试验的产量作为田间肥力的函数，以及检查孕妇特定血酶对预测未来癌症发病率的作用。

Proposal ID: DMS-0203243PI: EubankTitle: Spline smoothing and nonparametric regressionABSTRACT A number of nonparametric regression type problems are investigated. These problems are connected through the use of spline smoothing in their solution methodology. Specific problems that are studied include: 1) testing the lack-of-fit of a parametric regression model using a spline smoother in a setting where standard smoothing parameter consistency asymptotics do not hold under the null hypothesis, 2) estimation using spline smoothers in varying coefficient models, 3) variance estimation and testing for heteroscedasticity for partially linear models, 4) computational methods for nonlinear spline smoothing problems with both linear and nonlinear parameters, 5) adaptive selection of regularization parameters for spline smoothing of data from ill-posed integral equations and 6) computation and large sample properties of equality constrained local polynomial smoothers with applications to copula density estimation.The problems that are investigated in this research project concern regression analysis which represents the standard statistical approach to studying relationships between variables. The classical approach to regression analysis assumes that the form of the relationship between a collection of variables is known apart from a few unknown parameters that must be estimated from the data. This project uses more modern techniques that employ flexible or nonparametric curve fitting methods to produce estimators as well as to assess the validity of parametric models. New estimation methodologies are developed for several settings which include time varying coefficient models and partially linear models. Time varying coefficient models provide a generalization of parametric models where the parameters in the regression relationship are allowed to evolve as a function of some other variable such as time. This type of model is useful in a number of settings such as for analyzing data from longitudinal case studies and for prediction of lottery sales as a function of jackpot level. Partially linear models provide a mix of parametric and nonparametric methods where the regression relationships for some of the variables can be modeled parametrically while others must be handled using flexible nonparametric techniques. This latter type of model has been found useful, for example, in modeling yield from agricultural field trials as a function of field fertility and for examining the utility of particular blood enzymes in pregnant women for prediction of future incidences of cancer.