Extended Linear Models, Multivariate Splines and Anova

Extended Linear Models, Multivariate Splines and ANOVA by Mark Henry Hansen Doctor of Philosophy in Statistics University of California at Berkeley Professor Charles J. Stone, Chair In this dissertation, we pursue a theoretical investigation into several aspects of multivariate function estimation. In general, we con ne ourselves to estimators that are smooth, piecewise polynomial functions, or splines. In the last decade, a considerable body of literature on multivariate spline spaces has been amassed by approximation theorists, numerical analysts and computer scientists, and we hope to demonstrate the practicality of these tools for statistical applications. Initially, we consider estimating a regression function through ordinary least squares projections into certain spaces of splines. In order to tame the curse of dimensionality, we consider ANOVA decompositions of various function spaces. Finally, to accomodate more general estimation problems, we introduce the notion of an extended linear model and corresponding ANOVA decompositions. In the contexts of both regression and the extended linear model, the emphasis here is on rates of convergence.