Extended Linear Modeling with Splines

by Jianhua Z.\ Huang and Charles J. Stone

Abstract:

Extended linear models form a very general framework for
statistical modeling. Many practically important contexts
fit into this framework, including regression, logistic and
Poisson regression, density estimation, spectral density
estimation, and conditional density estimation. Moreover,
hazard regression, proportional hazard regression, marked
point process regression, and diffusion processes, 
all perhaps with time-dependent covariates,
also fit into this framework. Polynomial splines and their
tensor products provide a universal tool for constructing
maximum likelihood estimates for extended linear models. 
The theory of rates of convergence for such estimates as it 
applies both to fixed knot splines and to free knot splines 
will be surveyed, and the implications of this theory for 
the development of corresponding methodology will be 
discussed briefly.