Mathematical Sciences: Topics in Nonparametric Analysis and Model Building

建议解决如何将直观和简明的线性模型概念和技术扩展到非参数设置的问题，还建议开发可用于降维的非参数模型诊断技术，并解决特定模型的充分性问题。更准确地说，建议将重点放在与回归系数、相关系数、方差分析分解和主成分等常用线性模型思想相对应的程序上。此外，建议考虑用于建模、识别和诊断的非参数工具，包括线性、部分线性、可加性等测试。所有要考虑的估计器取决于曲线和曲面估计所需的平滑参数。这项研究的很大一部分将解决开发可靠的基于数据的方法来平滑参数选择的问题。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，开发可用于研究变量之间的关系并构建不那么依赖于特定假设的模型的一般模型、概念和过程已变得越来越可取和可能。建议将常用的线性模型思想和技术推广到这一更一般的情况，并评估结构更简单的模型的充分性。

It is proposed to address the question of how intuitive and concise linear model concepts and techniques can be extended to nonparametric settings and it is also proposed to develop nonparametric techniques for model diagnostics that can be used for dimensionality reduction and to address the question of adequacy of particular models. More precisely, it is proposed to focus on procedures that are counterparts to such commonly used linear model ideas as regression coefficients, correlation coefficients, ANOVA decompositions, and principal components. In addition, it is proposed to consider nonparametric tools for model building, identification, and diagnostics including tests for linearity, partial linearity, additivity, etc. All estimators to be considered depend on smoothing parameters needed in estimation of curves and surfaces. A large part of the research will address the problem of developing reliable data-based methods for smoothing parameter selection. With the advent of computer data bases of unprecedented size and complexity and with dramatic increase in computer power, it has become increasingly more desirable and possible to develop general models, concepts, and procedures that can be used to study relationships between variables and to construct models less dependent on specific assumptions. It is proposed to extend commonly used linear model ideas and techniques to this more general setting and to assess the adequacy of models with simpler structure.