On the Validity of the Bootstrap in Non-Parametric Functional Regression

Abstract

We consider the functional non-parametric regression model Y = r(chi) + epsilon, where the response Y is univariate, chi is a functional covariate (i.e. valued in some infinite-dimensional space), and the error epsilon satisfies E(epsilon vertical bar chi) = 0. For this model, the pointwise asymptotic normality of a kernel estimator (r) over cap(.) of r (.) has been proved in the literature. To use this result for building pointwise confidence intervals for r (.), the asymptotic variance and bias of (r) over cap(.) need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of (r) over cap(.) - r (.). Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.