Mathematical Sciences: Topics in Nonparametric Analysis and Model Building

DMS 9625777 Doksum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。考虑了回归系数、相关系数、决定系数和主成分等常用线性模型思想的非参数对应。该研究还研究了用于模型诊断的非参数技术，这些技术可用于降维并解决特定模型的充分性问题。研究了渐近样本和有限样本的性质，提出了基于数据的泛函和曲线平滑参数选择的可靠方法。随着前所未有的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、概念和过程已变得越来越可取和可能，这些模型、概念和过程可用于研究变量之间的关系并构建模型，而不依赖于僵硬的全局假设。最近在统计方面的许多工作都解决了对更一般和更灵活方法的这一需要。这项研究进一步扩展了这项工作，特别关注与许多常用的线性模型概念相对应的过程，并使用直观和熟悉的想法揭示数据中的重要特征。***

DMS 9625777 Doksum 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 has 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. ***