  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.
