Functional prediction through averaging estimated functional linear regression models

SummaryPrediction is often the primary goal of data analysis. In this work, we propose a novel model averaging approach to the prediction of a functional response variable. We develop a crossvalidation model averaging estimator based on functional linear regression models in which the response and the covariate are both treated as random functions. We show that the weights chosen by the method are asymptotically optimal in the sense that the squared error loss of the predicted function is as small as that of the infeasible best possible averaged function. When the true regression relationship belongs to the set of candidate functional linear regression models, the averaged estimator converges to the true model and can estimate the regression parameter functions at the same rate as under the true model. Monte Carlo studies and a data example indicate that in most cases the approach performs better than model selection.