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：针对函数型数据的均值与协方差函数，本文提出了非参数估计方法。该框架涵盖广泛的实践场景：随机轨迹既非必然可微，亦具有未知的正则性，并在离散设计点处伴随测量误差进行观测。测量误差可呈现异方差性，而设计点可随机生成或对所有曲线共享。估计量依赖于生成函数型数据的随机过程的局部正则性。我们利用函数型数据的复制与正则化特性，提出该局部正则性的简单估计量。继而采用“先平滑后估计”策略处理均值与协方差函数。该方法可应用于稀疏或密集采样曲线，计算简便且易于更新，在仿真实验中表现优异。基于实际数据集的仿真结果验证了新方法的有效性。