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：结合样本内与样本间的信息，我们在函数型数据框架下提出一种用于估计曲面局部正则性的简便方法。这些独立生成的曲面在可能随机的离散时间点上被带有误差地测量。我们推导了正则性估计量浓度的非渐近指数界，提出了一种各向异性指标并给出了该指标风险的指数界。本文提出两个应用：首先考虑具有区域变形的多分形二维布朗片类别，研究变形的非参数估计；其次，构建用于曲面重建的双变量核估计量并证明其达到极小化最优性。