Deep Regression for Repeated Measurements

Nonparametric mean function regression with repeated measurements serves as a cornerstone for many statistical branches, such as longitudinal/panel/functional data analysis. In this work, we investigate this problem using fully connected deep neural network (DNN) estimators with flexible shapes. A novel theoretical framework allowing arbitrary sampling frequency is established by adopting empirical process techniques to tackle clustered dependence. We then consider the DNN estimators for Hölder target function and illustrate a key phenomenon, the phase transition in the convergence rate, inherent to repeated measurements and its connection to the curse of dimensionality. Furthermore, we study several examples with low intrinsic dimensions, including the hierarchical composition model, low-dimensional support set and anisotropic Hölder smoothness. We also obtain new approximation results and matching lower bounds to demonstrate the adaptivity of the DNN estimators for circumventing the curse of dimensionality. Simulations and real data examples are provided to support our theoretical findings and practical implications. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.带重复测量的非参数均值函数回归，是纵向数据分析、面板数据分析与函数型数据分析等诸多统计学分支的核心基石。本研究采用具有灵活形态的全连接深度神经网络（DNN）估计器，对该问题展开探讨。通过引入经验过程技术处理聚类相依性，我们构建了可适配任意采样频率的全新理论框架。随后，我们针对赫尔德目标函数研究DNN估计器，并阐释了重复测量问题固有的关键现象：收敛速率相变，及其与维数灾难的内在关联。此外，我们针对多个低本征维度场景开展研究，涵盖分层组合模型、低维支撑集与各向异性赫尔德光滑性等情形。我们还得到了全新的逼近结果与匹配下界，用以证明DNN估计器在规避维数灾难方面的适应性。本研究通过仿真实验与真实数据案例，验证了理论发现与实际应用价值。本文的补充材料可在线获取，其中包含可复现本研究工作的标准化材料说明。

搜集汇总

背景与挑战

背景概述

该数据集研究了使用深度神经网络进行非参数均值函数回归的方法，特别关注重复测量数据的处理，并提供了理论框架、模拟和实际数据示例以支持研究发现。