Functional Index Model Research Articles

Cross-validated estimations in the single-functional index model

The functional index model consists in assuming that a functional explanatory variable acts on a scalar response only through its projection on one functional direction. The main aim of this work consists in estimating the unknown link function and the unknown functional index appearing in such a model. To this end, one focuses on an original cross-validation procedure allowing both estimations to be carried out simultaneously. This cross-validated method has several advantages: optimality property with respect to quadratic distances, ease of implementation and large flexibility for fitting and predicting purposes. One emphasizes the good behaviour in practice of the method. Finally, one discusses how such a single-functional index model can also be seen as a way of computing adaptative semi-metrics in the general frame of nonparametric functional regression.

Choix optimal du paramètre fonctionnel dans le modèle à indice fonctionnel simple

This Note focuses on an automatical choice of the functional parameter introduced throughout the single functional index model when one regresses a real r.v. on a functional r.v. (i.e. valued in an infinite-dimensional space). This Note aims to prove that a cross-validation-based method leads to an asymptotic optimal choice of the functional index. In addition, one states that such a choice for the functional index is a consistent estimator. To cite this article: A. Ait-Saïdi et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Modèle à indice fonctionnel simple

This paper concerns a generalization of the “Single Index Model” when the explanatory variable is valued in an infinite dimensional space. Such model will be called the “Single Functional Index Model”. The main contribution of this study is to propose a functional framework which includes the classical vectorial case. Pointwise asymptotic properties are stated under less restrictive conditions on the law of the explanatory variable than what it is usually assumed in the vectorial case. To cite this article: F. Ferraty et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).