SPLINE ESTIMATORS FOR THE FUNCTIONAL LINEAR MODEL

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 R. Many fields of appli- cations 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 con- tinuous 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