Statistica Sinica 13(2003), 571-591

SPLINE ESTIMATORS FOR THE FUNCTIONAL

LINEAR MODEL

Hervé Cardot$^\dagger$, Frédéric Ferraty$^\ddagger$ and Pascal Sarda$^\ddagger$

$^\dagger$INRA, Toulouse and $^\ddagger$Université Paul Sabatier, Toulouse

Abstract: We consider a regression setting where the response is a scalar and the predictor is a random function defined on a compact set of ${\Bbb R}$. Many fields of applications are concerned with this kind of data, for instance chemometrics when the predictor is a signal digitized in many points. Then, people have mainly considered the multivariate linear model and have adapted the least squares procedure to take care of highly correlated predictors. Another point of view is to introduce a continuous version of this model, i.e., the functional linear model with scalar response. We are then faced with the estimation of a functional coefficient or, equivalently, of a linear functional. We first study an estimator based on a B-splines expansion of the functional coefficient which in some way generalizes ridge regression. We derive an upper bound for the $L^2$ rate of convergence of this estimator. As an alternative we also introduce a smooth version of functional principal components regression for which $L^2$ convergence is achieved. Finally both methods are compared by means of a simulation study.

Key words and phrases: Convergence, functional linear model, Hilbert space valued random variables, principal components regression, regularization, splines.