Collaborative Research: Nonparametric Smoothing for Data with Multiple Components

几十年来，非参数平滑已成为许多经典统计问题的标准工具，这在一定程度上是由于计算能力的激增。相对较少的工作涉及更复杂的环境中的非参数平滑，其中数据具有多个分量，并且分析需要来自不同统计领域的技术的非平凡集成。本课题针对实际中常见的三类复杂数据，在光滑样条单因素方差分析模型的框架下，提出了一套非参数统计模型。现有的处理这三类数据的方法主要是参数和半参数的，其实际应用受到对响应对预测值或协变量的依赖结构的强烈假设的限制。本项目中提出的非参数方法结合了非参数高斯回归、非参数Logistic回归和非参数危险率估计，提供了更大的灵活性，在研究人员不确定依赖模式的探索阶段非常有用。与所提出的模型相伴随的是有用的推理工具，如模型选择和可信区间。结合非参数光滑样条、半参数估计和测量误差模型的渐近分析技术，研究了估计的渐近性质。当今联邦政府面临的重大挑战，如医疗改革、教育改革和金融体系改善，为数据提供了复杂的结构，需要准确、信息丰富和灵活的数据分析方法。提出的非参数平滑方法为开发适合于应对这些挑战的分析工具提供了一个创新的方向。这些类型的数据还可以在广泛的科学领域找到，如生物科学、经济学、社会科学、心理科学和生物医学研究。这项研究将促进相关统计领域的研究生和本科生的教育和培训。

Over the decades, nonparametric smoothing has become a standard tool for many classical statistical problems owing partly to the boom of computing power. Relatively little work has addressed nonparametric smoothing in more complex settings where data have multiple components and the analysis requires nontrivial integration of techniques from different statistical domains. This project concerns three types of such complex data that are common in practice, and propose a suite of nonparametric statistical models in the framework of smoothing spline ANOVA models. Existing methods for these three types of data are mainly parametric and semi-parametric, whose practical uses are limited by their strong assumptions on the dependence structure of response on predictors or covariates. The nonparametric methods proposed in this project, combining nonparametric Gaussian regression, nonparametric logistic regression and nonparametric hazard rate estimation, offer much more flexibility and are extremely useful at the exploratory stage when researchers are not certain of the pattern of dependence. Accompanied with the proposed models are useful inference tools such as model selection and confidence intervals. Asymptotic properties of the estimates are investigated through a combination of asymptotic analysis techniques for nonparametric smoothing splines, semiparametric estimation, and measurement error models. Major challenges to today's federal government, such as health care reform, education reform and financial system improvement, provide data with complex structures that call for accurate, informative and flexible data analysis methods. The proposed nonparametric smoothing methods provide an innovative direction for developing analysis tools appropriate for tackling these challenges. These types of data can also be found in a broad spectrum of scientific fields such as biological sciences, economics, social sciences, psychological sciences, and biomedical studies. This research will advance education and training of graduate and undergraduate students in the relevant statistical areas.