  We consider nonparametric estimation of the mean and covariance functions for
functional/longitudinal data. Strong uniform convergence rates are developed
for estimators that are local-linear smoothers. Our results are obtained in a
unified framework in which the number of observations within each curve/cluster
can be of any rate relative to the sample size. We show that the convergence
rates for the procedures depend on both the number of sample curves and the
number of observations on each curve. For sparse functional data, these rates
are equivalent to the optimal rates in nonparametric regression. For dense
functional data, root-n rates of convergence can be achieved with proper
choices of bandwidths. We further derive almost sure rates of convergence for
principal component analysis using the estimated covariance function. The
results are illustrated with simulation studies.
