Spline Smoothing and Nonparametric Regression

样条平滑与非参数回归摘要研究了一类非参数回归问题。这些问题通过在求解方法中使用样条平滑来联系起来。研究的具体问题包括：1)在零假设下标准平滑参数一致性渐近不成立的情况下，使用样条平滑检验参数回归模型的非拟合性；2)在变系数模型中使用样条平滑进行估计；3)部分线性模型的方差估计和异方差检验；4)线性和非线性参数非线性样条平滑问题的计算方法。5)不适定积分方程数据样条平滑的正则化参数自适应选择；6)等式约束局部多项式平滑的计算和大样本性质及其在耦合密度估计中的应用。本研究项目所研究的问题涉及回归分析，这是研究变量之间关系的标准统计方法。经典的回归分析方法假设，除了一些必须从数据中估计的未知参数之外，一组变量之间的关系的形式是已知的。本项目使用更现代的技术，采用灵活或非参数曲线拟合方法来产生估计器以及评估参数模型的有效性。针对时变系数模型和部分线性模型，提出了新的估计方法。时变系数模型提供了参数模型的泛化，其中回归关系中的参数可以作为其他变量（如时间）的函数而演变。这种类型的模型在许多情况下都很有用，比如分析纵向案例研究的数据，以及预测彩票销售作为头奖水平的函数。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化建模，而其他变量必须使用灵活的非参数技术来处理。人们发现后一种模型很有用，例如，可以将农业田间试验的产量作为田间肥力的函数进行建模，也可以检查孕妇体内特定血液酶的效用，以预测未来的癌症发病率。

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.