On the effect of noisy measurements of the regressor in functional linear models

We consider the estimation of the slope function in functional linear regression, where a scalar response Y is modelled in dependence of a random function X, when Y and only a panel Z ₁,…,Z _L of noisy measurements of X are observable. Assuming an i.i.d. sample of (Y,Z ₁,…,Z _L ) of size n we propose an estimator of the slope which is based on a dimension reduction technique and additional thresholding. We derive in terms of both the sample size n and the panel size L a lower bound of a maximal weighted risk over a certain ellipsoid of slope functions and a certain class of covariance operators associated with the regressor X. It is shown that the proposed estimator attains this lower bound up to a constant and hence it is minimax-optimal. The results are illustrated considering different configurations which cover in particular the estimation of the slope as well as its derivatives.