  We consider the estimation of the slope function in functional linear
regression, where scalar responses are modeled in dependence of random
functions. Cardot and Johannes [J. Multivariate Anal. 101 (2010) 395-408] have
shown that a thresholded projection estimator can attain up to a constant
minimax-rates of convergence in a general framework which allows us to cover
the prediction problem with respect to the mean squared prediction error as
well as the estimation of the slope function and its derivatives. This
estimation procedure, however, requires an optimal choice of a tuning parameter
with regard to certain characteristics of the slope function and the covariance
operator associated with the functional regressor. As this information is
usually inaccessible in practice, we investigate a fully data-driven choice of
the tuning parameter which combines model selection and Lepski's method. It is
inspired by the recent work of Goldenshluger and Lepski [Ann. Statist. 39
(2011) 1608-1632]. The tuning parameter is selected as minimizer of a
stochastic penalized contrast function imitating Lepski's method among a random
collection of admissible values. This choice of the tuning parameter depends
only on the data and we show that within the general framework the resulting
data-driven thresholded projection estimator can attain minimax-rates up to a
constant over a variety of classes of slope functions and covariance operators.
The results are illustrated considering different configurations which cover in
particular the prediction problem as well as the estimation of the slope and
its derivatives. A simulation study shows the reasonable performance of the
fully data-driven estimation procedure.
