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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.

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