Nonparametric estimators for the mean and the covariance functions of functional data are proposed. The setup covers a wide range of practical situations. The random trajectories are, not necessarily differentiable, have unknown regularity, and are measured with error at discrete design points. The measurement error could be heteroscedastic. The design points could be either randomly drawn or common for all curves. The estimators depend on the local regularity of the stochastic process generating the functional data. We consider a simple estimator of this local regularity which exploits the replication and regularization features of functional data. Next, we use the ``smoothing first, then estimate'' approach for the mean and the covariance functions. They can be applied with both sparsely or densely sampled curves, are easy to calculate and to update, and perform well in simulations. Simulations built upon an example of real data set, illustrate the effectiveness of the new approach.

翻译：本文提出了函数型数据均值与协方差函数的非参数估计方法。该框架适用于广泛的实际情况：随机轨迹不一定可微，具有未知的规律性，且在离散设计点上存在测量误差；测量误差可具有异方差性；设计点既可以是随机抽取的，也可以对所有曲线保持相同。所提出的估计量依赖于生成函数型数据的随机过程的局部规律性。我们利用函数型数据的重复性与正则化特性，构建了一种简单的局部规律性估计量。随后，采用“先平滑，后估计”的方法处理均值与协方差函数。该方法适用于稀疏或密集采样的曲线，计算与更新简便，在仿真实验中表现良好。基于真实数据集的仿真实验验证了新方法的有效性。