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A semiparametric model for Hilbertian random variables. (Un modèle semi-paramétrique pour variables aléatoires hilbertiennes.) (French. Abridged English version) Zbl 0996.62035
Summary: 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 K.-C. Li ’s sliced inverssse regression [J. Am. Stat. Assoc. 86, No. 414, 316-342 (1991; Zbl 0742.62044)]. However, in the Hilbertian context, SIR requires some adaptations of the estimation procedure and results concerning the consistency of the proposed estimates are given.

MSC:
62G08 Nonparametric regression and quantile regression
46N30 Applications of functional analysis in probability theory and statistics
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