SMOOTHING SPLINES ESTIMATORS FOR FUNCTIONAL LINEAR REGRESSION

• 成果类型: Article
• 署名作者: Crambes, Christophe; Kneip, Alois; Sarda, Pascal
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Federale Toulouse Midi-Pyrenees (ComUE); Institut National des Sciences Appliquees de Toulouse; University of Bonn
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/07-AOS563
• 发表日期: 2009
• 页码: 35-72

摘要

The paper considers functional linear regression, where scalar responses Y-1, ... , Y-n are modeled in dependence of random functions X-1, ... , X-n. We propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty. Theoretical analysis concentrates on the error in all out-of-sample prediction of the response for a new random function Xn+1. It is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the Structure Of the predictors. We then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves. For the case of models with errors-in-variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves. The methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day.