Advances on nonparametric regression for functional variables

TL;DR

This paper advances nonparametric kernel methods for predicting real variables from functional data, providing detailed asymptotic analysis and practical estimation techniques for constants involved.

Contribution

It offers a comprehensive asymptotic study of kernel-based nonparametric regression for functional variables, including convergence rates and distributional results.

Findings

Derived mean squared convergence rates with explicit constants

Established asymptotic distribution of the estimator

Discussed practical estimation and bootstrap methods for constants

Abstract

We consider the problem of predicting a real random variable from a functional explanatory variable. The problem is attacked by mean of nonparametric kernel approach which has been recently adapted to this functional context. We derive theoretical results by giving a deep asymptotic study of the behaviour of the estimate, including mean squared convergence (with rates and precise evaluation of the constant terms) as well as asymptotic distribution. Practical use of these results are relying on the ability to estimate these constants. Some perspectives in this direction are discussed including the presentation of a functional version of bootstrapping ideas.