Additive regression with Hilbertian responses

Abstract

This paper develops a foundation of methodology and theory for the estimation of structured nonparametric regression models with Hilbertian responses. Our method and theory are focused on the additive model, while the main ideas may be adapted to other structured models. For this, the notion of Bochner integration is introduced for Banach-space-valued maps as a generalization of Lebesgue integration. Several statistical properties of Bochner integrals, relevant for our method and theory and also of importance in their own right, are presented for the first time. Our theory is complete. The existence of our estimators and the convergence of a practical algorithm that evaluates the estimators are established. These results are nonasymptotic as well as asymptotic. Furthermore, it is proved that the estimators achieve the univariate rates in pointwise, $L^{2}$ and uniform convergence, and that the estimators of the component maps converge jointly in distribution to Gaussian random elements. Our numerical examples include the cases of functional, density-valued and simplex-valued responses, demonstrating the validity of our approach.