Deep Regression for Repeated Measurements
研究了用深度神经网络估计重复测量数据的非参数均值函数，建立了允许任意采样频率的理论框架，发现了收敛速度的相变现象，并展示了DNN估计器如何规避维度灾难。
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.