Many modern statistical applications involve a two-level sampling scheme that first samples subjects from a population and then samples observations on each subject. These schemes often are designed to learn both the population-level functional structures shared by the subjects and the functional characteristics specific to individual subjects. Common wisdom suggests that learning population-level structures benefits from sampling more subjects whereas learning subject-specific structures benefits from deeper sampling within each subject. Oftentimes these two objectives compete for limited sampling resources, which raises the question of how to optimally sample at the two levels. We quantify such sampling-depth trade-offs by establishing the $L_2$ minimax risk rates for learning the population-level and subject-specific structures under a hierarchical Gaussian process model framework where we consider a Bayesian and a frequentist perspective on the unknown population-level structure. These rates provide general lessons for designing two-level sampling schemes. Interestingly, subject-specific learning occasionally benefits more by sampling more subjects than by deeper within-subject sampling. We also construct estimators that adapt to unknown smoothness and achieve the corresponding minimax rates. We conduct two simulation experiments validating our theory and illustrating the sampling trade-off in practice, and apply these estimators to two real datasets.

翻译：许多现代统计应用涉及两级采样方案：首先从总体中抽取受试对象，然后对每个受试对象进行多次观测。这类方案通常旨在同时学习受试对象共享的总体层面函数结构以及个体受试对象的特定函数特征。传统观点认为，学习总体层面结构需要增加受试对象数量，而学习个体层面结构则需要对每个受试对象进行更深层次的采样。然而，这两个目标往往在有限采样资源下相互竞争，从而引发如何优化两级采样的关键问题。本文通过建立层级高斯过程模型框架下学习总体层面和个体层面结构的$L_2$极小极大风险率，量化了此类采样深度权衡问题。在该框架中，我们分别从贝叶斯和频率学派视角处理未知的总体层面结构。这些风险率可为两级采样方案的设计提供普适性指导。值得注意的是，在特定条件下，通过增加受试对象数量而非加深个体采样深度，反而更有利于个体层面结构的学习。我们还构建了能自适应未知光滑性并达到相应极小极大风险率的估计量。通过两组仿真实验验证理论结果并展示实际中的采样权衡，同时将所提估计量应用于两个真实数据集。