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.

中文翻译：

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希尔伯特响应的加性回归

本文为基于希尔伯特响应的结构化非参数回归模型的估计奠定了方法论和理论基础。我们的方法和理论侧重于加性模型，而主要思想可能适用于其他结构化模型。为此，引入了Bochner积分的概念，作为Lebesgue积分的推广，用于Banach空间值地图。首次介绍了与我们的方法和理论相关并且本身具有重要性的Bochner积分的几种统计性质。我们的理论是完整的。建立了我们的估计量的存在以及对估计量进行评估的实用算法的收敛性。这些结果是非渐近的和渐近的。此外，证明了估计量可以逐点实现单变量率，$ L ^ {2} $和均匀收敛，并且分量映射的估计量在分布到高斯随机元素时共同收敛。我们的数值示例包括功能，密度值和单纯形值响应的情况，证明了我们方法的有效性。