On rate optimal local estimation in functional linear model

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. The theory in this paper covers in particular point-wise estimation as well as the estimation of averages of the slope parameter. We show a lower bound of the maximal mean squared error for any estimator over a certain ellipsoid of slope parameters. This bound is essentially determined by the representer of the linear functional and the mapping properties of the covariance operator associated to the random function. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent under mild assumptions and can attain the lower bound up to a constant under additional regularity conditions. As illustration we consider Sobolev ellipsoids and smoothing covariance operators.