Free knot splines in concave extended linear modeling

Abstract

Many problems of practical interest can be formulated as the estimation of a certain function such as a regression function, logistic or other generalized regression function, density function, conditional density function, hazard function, or conditional hazard function. Extended linear modeling provides a convenient framework for using polynomial splines and their tensor products in such function estimation problems. Huang (Statist. Sinica 11 (2001) 173) has given a general treatment of the rates of convergence of maximum likelihood estimation in the context of concave extended linear modeling. Here these results are generalized to let the approximation space used in the fitting procedure depend on a vector of parameters. More detailed treatments are given for density estimation and generalized regression (including ordinary regression) on the one hand and for approximation spaces whose components are suitably regular free knot splines and their tensor products on the other hand.