Bandwidth selection for functional time series prediction

We propose a method to select the bandwidth for functional time series prediction. The idea underlying this method is to calculate the empirical risk of prediction using past segments of the observed series and to select as value of the bandwidth for prediction the bandwidth which minimizes this risk. We prove an oracle bound for the proposed bandwidth estimator showing that it mimics, asymptotically, the value of the bandwidth which minimizes the unknown theoretical risk of prediction based on past segments. We illustrate the usefulness of the proposed estimator in finite sample situations by means of a small simulation study and compare the resulting predictions with those obtained by a leave-one-curve-out cross-validation estimator used in the literature.