Asymptotics for penalized additive B-spline regression

TL;DR

This paper develops the asymptotic theory for penalized spline estimators in bivariate additive models, focusing on convergence, bias, variance, and normality, with numerical validation.

Contribution

It provides the first comprehensive asymptotic analysis of penalized spline estimators obtained via backfitting in bivariate additive models.

Findings

Convergence and uniqueness of the backfitting algorithm are established.

Asymptotic bias and variance formulas are derived.

Estimator's asymptotic normality enables confidence interval construction.

Abstract

This paper is concerned with asymptotic theory for penalized spline estimator in bivariate additive model. The focus of this paper is put upon the penalized spline estimator obtained by the backfitting algorithm. The convergence of the algorithm as well as the uniqueness of its solution are shown. The asymptotic bias and variance of penalized spline estimator are derived by an efficient use of the asymptotic results for the penalized spline estimator in marginal univariate model. Asymptotic normality of estimator is also developed, by which an approximate confidence interval can be obtained. Some numerical experiments confirming theoretical results are provided.