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.
