A semiparametric model for Hilbertian random variables

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. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.