Convergence of Nonparametric Functional Regression Estimates with Functional Responses

TL;DR

This paper establishes almost sure convergence rates for nonparametric functional regression estimators with functional predictors and responses, considering weak dependence and complex functional data structures.

Contribution

It provides a unified theoretical analysis of convergence rates for Nadaraya-Watson and nearest neighbor estimators in functional regression with weakly dependent data.

Findings

Almost sure convergence rates derived for both estimators

Analysis accounts for weak dependence in functional data

Handles complex functional responses and predictors

Abstract

We consider nonparametric functional regression when both predictors and responses are functions. More specifically, we let $(X_1,Y_1),...,(X_n,Y_n)$ be random elements in $\mathcal{F}\times\mathcal{H}$ where $\mathcal{F}$ is a semi-metric space and $\mathcal{H}$ is a separable Hilbert space. Based on a recently introduced notion of weak dependence for functional data, we showed the almost sure convergence rates of both the Nadaraya-Watson estimator and the nearest neighbor estimator, in a unified manner. Several factors, including functional nature of the responses, the assumptions on the functional variables using the Orlicz norm and the desired generality on weakly dependent data, make the theoretical investigations more challenging and interesting.