Nonparametric models for functional data, with application in regression, time series prediction and curve discrimination

Abstract

The aim of this article is to investigate a new approach for estimating a regression model with scalar response and in which the explanatory variable is valued in some abstract semi-metric functional space. Nonparametric estimates are introduced, and their behaviors are investigated in the situation of dependent data. Our study contains asymptotic results with rates. The curse of dimensionality, which is of great importance in this infinite dimensional setting, is highlighted by our asymptotic results. Some ideas, based on fractal dimension modelizations, are given to reduce dimensionality of the problem. Generalization of the model leads to possible applications in several fields of applied statistics, and we present three applications among these namely: regression estimation, time-series prediction, and curve discrimination. As a by-product of our approach in the finite-dimensional context, we give a new proof for the rates of convergence of some Nadaraya–Watson kernel-type smoother without needing any smoothness assumption on the density function of the explanatory variables.

Keywords:

• Functional data
• Nonparametric regression
• Time-series prediction
• Curves discrimination
• Fractal dimension
• Semi-metric space

Acknowledgement

The authors would like to thank all the participants of the working group STAPH on Functional Statistics at the University Paul Sabatier of Toulouse, for their stimulating and continuous helpful comments. The activities of this group are available on http://www.lsp.ups-tlse.fr/Fp/Ferraty/staph.html.