The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions

TL;DR

This paper develops and validates bootstrap methods for kernel regression estimators in a complex setting where both response and predictor are functions, under stationary ergodic data, enabling better confidence set construction.

Contribution

It introduces a Gaussian limit law for a kernel estimator in double functional regression and proves the asymptotic validity of naive and wild bootstrap procedures in this context.

Findings

Gaussian limiting distribution for the kernel estimator

Asymptotic validity of bootstrap procedures

Applicable to spatial and ergodic functional data

Abstract

We consider the double functional nonparametric regression model $Y=r(X)+\epsilon$, where the response variable $Y$ is Hilbert space-valued and the covariate $X$ takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields indexed by the regular lattice $\mathbb{Z}^N$ for some $N\in\mathbb{N}_+$. We consider a kernel estimator of the Nadaraya--Watson type for the regression operator $r$ and study its limiting law which is a Gaussian operator on the Hilbert space. Moreover, we investigate both a naive and a wild bootstrap procedure in the double functional setting and demonstrate their asymptotic validity. This is quite useful as building confidence sets based on an asymptotic Gaussian distribution is often difficult.