Analysis of repeated measurements for a sample of subjects has been intensively studied with several important branches developed, including longitudinal/panel/functional data analysis, while nonparametric regression of the mean function serves as a cornerstone that many statistical models are built upon. In this work, we investigate this problem using fully connected deep neural network (DNN) estimators with flexible shapes. A comprehensive theoretical framework is established by adopting empirical process techniques to tackle clustered dependence. We then derive the nearly optimal convergence rate of the DNN estimators in H\"older smoothness space, and illustrate the phase transition phenomenon inherent to repeated measurements and its connection to the curse of dimensionality. Furthermore, we study the function spaces with low intrinsic dimensions, including the hierarchical composition model, anisotropic H\"older smoothness and low-dimensional support set, and also obtain new approximation results and matching lower bounds to demonstrate the adaptivity of the DNN estimators for circumventing the curse of dimensionality.

翻译：针对受试者样本重复测量数据的分析已得到深入研究，并形成了纵向/面板/函数型数据分析等多个重要分支，其中均值函数的非参数回归作为诸多统计模型的基础具有基石性地位。本文采用具有灵活结构的全连接深度神经网络（DNN）估计器研究该问题。通过引入经验过程技术处理聚类相依性，我们建立了完备的理论框架。在Hölder光滑空间中推导出DNN估计量的近最优收敛速率，揭示了重复测量数据固有的相变现象及其与维度灾难的关联。进一步，我们研究了低内在维度的函数空间，包括分层组合模型、各向异性Hölder光滑性和低维支撑集，获得了新的逼近结果和匹配下界，证明了DNN估计器在规避维度灾难方面的自适应性。