Un modèle semi-paramétrique pour variables aléatoires hilbertiennes

This Note deals with a semi-parametric model for Hilbertian random variables. The model is said semi-parametric by analogy with the finite dimensional case since the model involves a composition of any measurable mapping with a linear mapping which represents the “parametric” part. Under mild conditions, we derive a way for estimating this linear component in a particular case. We show that this method is actually a generalization of Li's Sliced Inverse Regression. However, in the Hilbertian context, SIR requires some adaptations of the estimation procedure and results concerning the consistency of the proposed estimates are given.