  Functional data analysis almost always involves smoothing discrete
observations into curves, because they are never observed in continuous time
and rarely without error. Although smoothing parameters affect the subsequent
inference, data-driven methods for selecting these parameters are not
well-developed, frustrated by the difficulty of using all the information
shared by curves while being computationally efficient. On the one hand,
smoothing individual curves in an isolated, albeit sophisticated way, ignores
useful signals present in other curves. On the other hand, bandwidth selection
by automatic procedures such as cross-validation after pooling all the curves
together quickly become computationally unfeasible due to the large number of
data points. In this paper we propose a new data-driven, adaptive kernel
smoothing, specifically tailored for functional principal components analysis
through the derivation of sharp, explicit risk bounds for the eigen-elements.
The minimization of these quadratic risk bounds provide refined, yet
computationally efficient bandwidth rules for each eigen-element separately.
Both common and independent design cases are allowed. Rates of convergence for
the estimators are derived. An extensive simulation study, designed in a
versatile manner to closely mimic the characteristics of real data sets
supports our methodological contribution. An illustration on a real data
application is provided.
