Mathematical Sciences: Semi-Parmetric Regression with Counting Data

本研究旨在建立具有计数资料的半参数回归模型中估计和关键量的分布性质，重点研究广义线性模型背景下的Poisson和二项数据。估计将被认为是半参数的，因为它有一个参数分量和一个受某种“平滑”约束的分量，通常在广义样条法的设置中。必须为这项研究开发的必要工具是将这项研究的结果传递给应用统计学家的有效计算方法，用于光滑的一般线性模型问题，其中设计点落在规则的多维网格上。统计分布理论方面的这项研究将提供有用的分析工具，直接转化为选择概率模型的图形程序和评估其充分性的诊断程序。这项研究针对的是计数数据的简单参数模型不能保证的情况。

This research is to establish the distributional properties of estimates and pivotal quantities in semi-parametric regression models with counting data, focusing on Poisson and binomial data in the setting of the generalized linear model. Estimates will be considered semi-parametric in the sense of having a parametric component and a component which is subject to some "smoothing" constraint, usually in the setting of generalized splines. Necessary tools that must be developed for this research and for transfering results from this research to the applied statisician are efficient computational methods for smooth general linear model problems in which the design points fall on a regular, multidimensional grid. This research in statistical distribution theory will provide useful analytical tools which translate directly into graphical procedures for selecting probability models and diagnostic procedures for assessing their adequacy. This research is directed toward situations where simple parametric models of counting data are not warranted.