In this paper, in a multivariate setting we derive near optimal rates of convergence in the minimax sense for estimating partial derivatives of the mean function for functional data observed under a fixed synchronous design over H\"older smoothness classes. We focus on the supremum norm since it corresponds to the visualisation of the estimation error, and is closely related to the construction of uniform confidence bands. In contrast to mean function estimation, for derivative estimation the smoothness of the paths of the processes is crucial for the rates of convergence. On the one hand, if the paths have higher-order smoothness than the order of the partial derivative to be estimated, the parametric $\sqrt n$ rate can be achieved under sufficiently dense design. On the other hand, for processes with rough paths of lower-order smoothness, we show that the rates of convergence are necessarily slower than the parametric rate, and determine a near-optimal rate at which estimation is still possible. We implement a multivariate local polynomial derivative estimator and illustrate its finite-sample performance in a simulation as well as for two real-data sets. To assess the smoothness of the sample paths in the applications we further discuss a method based on comparing restricted estimates of the partial derivatives of the covariance kernel.

翻译：本文在多元设定下，针对固定同步设计下观测的函数数据，在H\"older光滑性类上推导了均值函数偏导数估计在极小极大意义下的近最优收敛速率。我们重点关注上确界范数，因为它对应着估计误差的可视化，且与一致置信带的构建密切相关。与均值函数估计不同，对于导数估计而言，过程路径的光滑性对收敛速率至关重要。一方面，若路径的光滑性阶数高于待估计偏导数的阶数，则在足够密集的设计下可以达到参数性的$\sqrt n$速率。另一方面，对于具有较低阶光滑性的粗糙路径过程，我们证明其收敛速率必然慢于参数速率，并确定了仍可实现估计的近最优速率。我们实现了一种多元局部多项式导数估计器，并通过模拟和两个真实数据集展示了其有限样本性能。为评估应用中样本路径的光滑性，我们进一步讨论了一种基于协方差核偏导数受限估计比较的方法。