We consider the nonparametric estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. The theory in this presentation covers both the estimation of the slope function or its derivatives (global case) as well as the estimation of a linear functional of the slope function (local case). We propose an estimator of the slope function which is based on dimension reduction and additional thresholding. Moreover, replacing the unknown slope function by this estimator we obtain in the local case a plug-in estimator of the value of a linear functional evaluated at the slope. It is shown that in both the global and the local case these estimators can attain minimax optimal rates of convergence up to a constant. However, the estimator of the slope function 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 which combines model selection and Lepski’s method. It is inspired by the recent work of Goldenshluger and Lepski [2011]. It is shown that the adaptive estimator with data-driven choice of the dimension parameter can attain the lower minimax risk bound in the global case up to a constant and in the local case up to a logarithmic factor, and this over a variety of classes of slope functions and covariance operators.

H. Cardot and J. Johannes. Thresholding projection estimators in functional linear models. Journal of Multivariate Analysis, 101(2):395–408, 2010.
F. Comte and J. Johannes. Adaptive functional linear regression. Technical report, Université Paris Descartes, Laboratoire MAP5, 2011.
A. Goldenshluger and O. Lepski. Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality. The Annals of Statistics, 39:1608–1632, 2011.
J. Johannes and R. Schenk. On rate optimal local estimation in functional linear model. http://arxiv.org/abs/0902.0645, Université catholique de Louvain, 2011a.
J. Johannes and R. Schenk. Adaptive estimation of linear functionals in functional linear models. Technical report, Université catholique de Louvain, 2011b.