On rate optimal local estimation in functional linear model

We consider the problem of estimating for a given representer the value of a linear functional of the slope parameter in functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of random functions X1,...,Xn. The proposed estimators are based on dimension reduction and additional thresholding. The minimax optimal rate of convergence of the estimator is derived assuming that the slope parameter and the representer belong to some ellipsoid which are in a certain sense linked to the covariance operator associated to the regressor. We illustrate these results by considering Sobolev ellipsoids and finitely or infinitely smoothing covariance operator.