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

We consider the functional non-parametric regression model "Y"&equals; "r"(<b>"χ"</b>)&plus;"&epsiv;", where the response "Y" is univariate, <b>"χ"</b> is a functional covariate (i.e. valued in some infinite-dimensional space), and the error "&epsiv;" satisfies "E"("&epsiv;" | <b>"χ"</b>) &equals; 0. For this model, the pointwise asymptotic normality of a kernel estimator <formula format="inline"><file name="sjos_662_mu1.gif" type="gif" /></formula> 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 <formula format="inline"><file name="sjos_662_mu2.gif" type="gif" /></formula> 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 <formula format="inline"><file name="sjos_662_mu3.gif" type="gif" /></formula>. 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. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.