A General Approach for Asymptotics of Penalized Spline Estimation in Extended Linear Models

Abstract

The penalized spline estimator has been formally introduced in the context of the nonparametric regression model. Despite the wide range of its application, the theory of penalized spline estimator has fallen behind. In this dissertation, we first look into the existing theoretical results about the penalized spline estimator with some scrutiny and point out the room left for improvement, that is, they are built upon certain asymptotic scenarios for one specific model. Then we state and prove a unified theory, that is, the convergence rate of the penalized spline estimator is established for the set of extended linear models, which holds under various asymptotic scenarios. The application of the main theory to a list of extended linear models including nonparametric regression, generalized regression, counting process, density estimation, spectral density estimation, diffusion process and nonparametric M-regression is also provided for completeness.