We consider the estimation of the value of a linear functional of the slope
parameter in functional linear regression, where scalar responses are modeled
in dependence of random functions. In Johannes and Schenk [2010] it has been
shown that a plug-in estimator based on dimension reduction and additional
thresholding can attain minimax optimal rates of convergence up to a constant.
However, this estimation procedure 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 these are
unknown in practice, we investigate a fully data-driven choice of the tuning
parameter based on a combination of model selection and Lepski's method, which
is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning
parameter is selected as the minimizer of a stochastic penalized contrast
function imitating Lepski's method among a random collection of admissible
values. We show that this adaptive procedure attains the lower bound for the
minimax risk up to a logarithmic factor over a wide range of classes of slope
functions and covariance operators. In particular, our theory covers point-wise
estimation as well as the estimation of local averages of the slope parameter
