Asymptotic properties of penalized spline estimators

• 成果类型: Article
• 署名作者: Claeskens, Gerda; Krivobokova, Tatyana; Opsomer, Jean D.
• 署名单位: KU Leuven; KU Leuven; University of Gottingen; Colorado State University System; Colorado State University Fort Collins
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asp035
• 发表日期: 2009
• 页码: 529544
• 关键词: smoothing noisy data; local asymptotics; regression; CONVERGENCE; rates; constraints; matrices; error

摘要

We study the class of penalized spline estimators, which enjoy similarities to both regression splines, without penalty and with fewer knots than data points, and smoothing splines, with knots equal to the data points and a penalty controlling the roughness of the fit. Depending on the number of knots, sample size and penalty, we show that the theoretical properties of penalized regression spline estimators are either similar to those of regression splines or to those of smoothing splines, with a clear breakpoint distinguishing the cases. We prove that using fewer knots results in better asymptotic rates than when using a large number of knots. We obtain expressions for bias and variance and asymptotic rates for the number of knots and penalty parameter.