Regression when both response and predictor are functions

We consider a nonparametric regression model where the response YY and the covariate XX are both functional (i.e. valued in some infinite-dimensional space). We define a kernel type estimator of the regression operator and we first establish its pointwise asymptotic normality. The double functional feature of the problem makes the formulas of the asymptotic bias and variance even harder to estimate than in more standard regression settings, and we propose to overcome this difficulty by using resampling ideas. Both a naive and a wild componentwise bootstrap procedure are studied, and their asymptotic validity is proved. These results are also extended to data-driven bases which is a key point for implementing this methodology. The theoretical advances are completed by some simulation studies showing both the practical feasibility of the method and the good behavior for finite sample sizes of the kernel estimator and of the bootstrap procedures to build functional pseudo-confidence area.