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February 2009 Smoothing splines estimators for functional linear regression
Christophe Crambes, Alois Kneip, Pascal Sarda
Ann. Statist. 37(1): 35-72 (February 2009). DOI: 10.1214/07-AOS563


The paper considers functional linear regression, where scalar responses Y1, …, Yn are modeled in dependence of random functions X1, …, Xn. 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 an 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.


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Christophe Crambes. Alois Kneip. Pascal Sarda. "Smoothing splines estimators for functional linear regression." Ann. Statist. 37 (1) 35 - 72, February 2009.


Published: February 2009
First available in Project Euclid: 16 January 2009

zbMATH: 1169.62027
MathSciNet: MR2488344
Digital Object Identifier: 10.1214/07-AOS563

Primary: 62G05 , 62G20
Secondary: 60G12 , 62M20

Keywords: Functional linear regression , functional parameter , functional variable , smoothing splines

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • February 2009
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