Smooth and Rough Sample Paths in Mean Derivative Estimation for Functional Data

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 Hölder smoothness classes. In contrast to mean function estimation, for derivative estimation, the smoothness of the sample paths of the processes is crucial. For processes with rough sample paths of lower‐order smoothness than the order of the partial derivative to be estimated, we determine a novel, slower than parametric optimal rate of convergence. For processes with sample paths of higher‐order smoothness, we show that the parametric n$$ \sqrt{n} $$‐rate can still be achieved under sufficiently dense design. We conduct our analysis in the supremum norm since it corresponds to the visualization of the estimation error. Further, as a basis for the construction of uniform confidence bands, we also derive a central limit theorem in the space of continuous functions equipped with the sup‐norm. Derivative estimation is of quite some interest in functional data analysis, for example, to assess the dynamics of the underlying processes. 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 determine 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.