2009/01/16 by Christophe Crambes, Alois Kneip, Pascal Sarda
Mathematics · #Advanced Statistical Methods and Models #Covariance #Covariance matrix #Estimator #Function (biology) #Functional data analysis #Minimax #Smoothing #Smoothing spline #Smoothness #Spline (mechanical) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #math.ST #msc:60G12 #msc:62G05 #msc:62G20 #msc:62M20 #stat.TH
paper · pdf · doi:10.1214/07-aos563
published as Annals of Statistics 2009, Vol. 37, No. 1, 35-72 · Published in at http://dx.doi.org/10.1214/07-AOS563 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2009/01/16 · arxiv created 2009/02/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.