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层采样方案：首先从总体中抽取受试者，然后对每个受试者进行观测采样。这类方案通常旨在学习受试者共有的总体层函数结构以及单个受试者特有的函数特征。传统观点认为，学习总体层结构需要增加受试者数量，而学习受试者特有结构则需对每个受试者进行更深层采样。这两种目标往往在有限采样资源下相互竞争，由此引发如何优化两层采样的核心问题。我们通过建立分层高斯过程模型框架下学习总体层与受试者特有结构的$L_2$极小极大风险率来量化这种采样深度权衡，其中分别从贝叶斯和频率学派视角处理未知的总体层结构。这些风险率为设计两层采样方案提供了普适性指导原则。值得注意的是，在特定条件下，通过增加受试者数量比加深受试者内采样更有利于学习个体特有结构。我们还构建了能自适应未知光滑性并达到相应极小极大率的估计量。通过两项仿真实验验证理论并展示实际采样权衡，同时将这些估计量应用于两个真实数据集。