Asymptotics for Penalized Additive <i>B</i>-spline Regression

Abstract

This paper is concerned with the asymptotic theory for penalized spline estimators in additive models. The focus of this paper is on the penalized spline estimators obtained by the backfitting algorithm. The convergence of the algorithm as well as the uniqueness of its solution are shown. Asymptotic equivalence between the penalized spline estimators by the backfitting algorithm and the convenient estimators proposed by Marx and Eilers (1998) is addressed. Asymptotic normality of the estimators is also developed, by which an approximate confidence interval can be obtained. Some numerical experiments confirming theoretical results are provided.