A number of nonparametric regression type problems are investigated. These problems are connected through the use of spline smoothing in their solution methodology. Specific problems that are studied include: 1) testing the lack-of-fit of a parametric regression model using a spline smoother in a setting where standard smoothing parameter consistency asymptotics do not hold under the null hypothesis, 2) estimation using spline smoothers in varying coefficient models, 3) variance estimation and testing for heteroscedasticity for partially linear models, 4) computational methods for nonlinear spline smoothing problems with both linear and nonlinear parameters, 5) adaptive selection of regularization parameters for spline smoothing of data from ill-posed integral equations and 6) computation and large sample properties of equality constrained local polynomial smoothers with applications to copula density estimation.
The problems that are investigated in this research project concern regression analysis which represents the standard statistical approach to studying relationships between variables. The classical approach to regression analysis assumes that the form of the relationship between a collection of variables is known apart from a few unknown parameters that must be estimated from the data. This project uses more modern techniques that employ flexible or nonparametric curve fitting methods to produce estimators as well as to assess the validity of parametric models. New estimation methodologies are developed for several settings which include time varying coefficient models and partially linear models. Time varying coefficient models provide a generalization of parametric models where the parameters in the regression relationship are allowed to evolve as a function of some other variable such as time. This type of model is useful in a number of settings such as for analyzing data from longitudinal case studies and for prediction of lottery sales as a function of jackpot level. Partially linear models provide a mix of parametric and nonparametric methods where the regression relationships for some of the variables can be modeled parametrically while others must be handled using flexible nonparametric techniques. This latter type of model has been found useful, for example, in modeling yield from agricultural field trials as a function of field fertility and for examining the utility of particular blood enzymes in pregnant women for prediction of future incidences of cancer.
