A unified framework for spline estimators

TL;DR

This paper introduces a comprehensive framework for analyzing the asymptotic behavior of all periodic spline estimators, unifying regression, penalized, and smoothing splines through explicit basis representations.

Contribution

It provides a unified approach to study the asymptotic properties of various spline estimators using explicit basis forms and kernel expressions.

Findings

Derived an explicit asymptotic equivalent kernel for all spline estimators.

Showed the bandwidth depends on knots and smoothing parameter.

Discussed strategies for optimal parameter selection.

Abstract

This article develops a unified framework to study the asymptotic properties of all periodic spline-based estimators, that is, of regression, penalized and smoothing splines. The explicit form of the periodic Demmler-Reinsch basis in terms of exponential splines allows the derivation of an expression for the asymptotic equivalent kernel on the real line for all spline estimators simultaneously. The corresponding bandwidth, which drives the asymptotic behavior of spline estimators, is shown to be a function of the number of knots and the smoothing parameter. Strategies for the selection of the optimal bandwidth and other model parameters are discussed.