Collaborative Research: Nonparametric smoothing for data with multiple components

几十年来，非参数平滑已经成为许多经典统计问题的标准工具，部分原因是计算能力的激增。相对较少的工作已经解决了非参数平滑在更复杂的设置中，数据有多个组件和分析需要从不同的统计域的技术的非平凡的集成。本计画针对三种在实务中常见的复杂资料，在平滑样条变异数分析模型的架构下，提出一套非参数统计模型。现有的方法，这三种类型的数据主要是参数和半参数，其实际用途是有限的，其强烈的假设依赖结构的预测或协变量的响应。在这个项目中提出的非参数方法，结合非参数高斯回归，非参数逻辑回归和非参数风险率估计，提供了更多的灵活性，是非常有用的探索阶段，当研究人员不确定的依赖模式。伴随着所提出的模型是有用的推理工具，如模型选择和置信区间。 通过非参数光滑样条，半参数估计和测量误差模型的渐近分析技术相结合的估计的渐近性质进行了研究。当今联邦政府面临的主要挑战，如医疗保健改革、教育改革和金融系统改进，提供了具有复杂结构的数据，这需要准确、信息丰富和灵活的数据分析方法。所提出的非参数平滑方法为开发适合应对这些挑战的分析工具提供了一个创新的方向。这些类型的数据也可以在广泛的科学领域中找到，如生物科学，经济学，社会科学，心理科学和生物医学研究。这项研究将促进对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.