Cross-validated estimations in the single-functional index model

Abstract

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.

Keywords:

• adaptative semi-metric
• cross-validation
• data-driven selection of functional index
• functional data analysis
• functional index model
• mean integrated squared error
• nonparametric statistics
• optimality properties

Acknowledgements

The authors thank all the participants of the working group STAPH (http://www.lsp.ups-tlse.fr/staph) on functional statistics at the University Paul Sabatier of Toulouse for their stimulating and helpful comments.

Notes

¹Let {V _n } _n∈ℕ be a random sequence: V _n =O _a.co.(u _n )⇔ ∃ ε₀>0, ∑ _n∈ℕ P(|V _n |>ε₀ u _n )<+∞ and by Borel–Cantelli's Lemma it holds that V _n =O _a.co.(u _n )↠V _n =O_a.s.(u _n ).