Topics in Nonparametric Analysis and Model Building

DMS 9626348 Samarov这项研究解决了如何将直观和简洁的线性模型概念和技术扩展到非参数设置的问题。 这些常用的线性模型的想法，如回归系数，相关系数，决定系数和主成分的非参数对应物被认为是。 该研究还研究了模型诊断的非参数技术，可用于降维和解决特定模型的充分性问题。 渐近和有限样本的性质进行了研究，并制定可靠的数据为基础的方法光滑的泛函和曲线的参数选择的问题。 随着空前规模和复杂性的计算机数据库的出现，以及计算机能力的急剧增加，开发更灵活的模型、概念和程序变得越来越合乎需要和可能，这些模型、概念和程序可用于研究变量之间的关系，并在不依赖于严格的全局假设的情况下构建模型。 最近在统计方面的许多工作都涉及到对更普遍和灵活的方法的需要。 这项研究进一步扩展了这项工作，特别关注与许多常用线性模型概念相对应的程序，并使用直观和熟悉的想法揭示数据中的重要特征。

DMS 9626348 Samarov This research addresses the question of how intuitive and concise linear model concepts and techniques can be extended to nonparametric settings. Nonparametric counterparts of such commonly used linear model ideas as regression coefficients, correlation coefficients, coefficient of determination, and principal components are considered. The research also studies nonparametric techniques for model diagnostics that can be used for dimensionality reduction and to address the question of adequacy of particular models. Both asymptotic and finite sample properties are studied, and the problem of developing reliable data-based methods for smoothing parameter selection for functionals and curves is addressed. With the advent of computer data bases of unprecedented size and complexity and with the dramatic increase in computer power, it as become increasingly more desirable and possible to develop more flexible models, concepts, and procedures that can be used to study relationships between variables and to construct models without relying on rigid global assumptions. Much of the recent work in statistics have addressed this need for more general and flexible methods. This research further extends this work with a special focus on procedures that are counterparts of many commonly used linear model concepts and that expose important features in the data using intuitive and familiar ideas.