Combining information both within and between sample realizations, we propose a simple estimator for the local regularity of surfaces in the functional data framework. The independently generated surfaces are measured with errors at possibly random discrete times. Non-asymptotic exponential bounds for the concentration of the regularity estimators are derived. An indicator for anisotropy is proposed and an exponential bound of its risk is derived. Two applications are proposed. We first consider the class of multi-fractional, bi-dimensional, Brownian sheets with domain deformation, and study the nonparametric estimation of the deformation. As a second application, we build minimax optimal, bivariate kernel estimators for the reconstruction of the surfaces.

翻译：结合样本内部和样本间的信息，本文提出了函数型数据框架下曲面局部正则性的一种简单估计量。这些独立生成的曲面在可能随机的离散时间点被带有误差地测量。我们推导了正则性估计量浓度的非渐近指数界。提出了各向异性的指标，并推导了其风险的指数界。本文提出了两个应用。首先，我们考虑具有区域变形的多分形二维布朗叶类，并研究变形的非参数估计。其次，我们构建了用于曲面重建的极小极大最优双变量核估计量。