We study the estimation of the mean function of a continuous-time stochastic process and
its derivatives. The covariance function of the process is assumed to be nonparametric and
to satisfy mild smoothness conditions. Assuming that n independent realizations
of the process are observed at a sampling design of size N generated by a positive
density, we derive the asymptotic bias and variance of the local polynomial estimator as
n,N
increase to infinity. We deduce optimal sampling densities, optimal bandwidths, and
propose a new plug-in bandwidth selection method. We establish the asymptotic performance
of the plug-in bandwidth estimator and we compare, in a simulation study, its performance
for finite sizes n,N to the cross-validation and the optimal
bandwidths. A software implementation of the plug-in method is available in the R
environment.