Free knot splines in concave extended linear modeling

Charles J. Stone and Jianhua Z. Huang

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 (2001) 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.