Variable selection for functional linear models with functional predictors and a functional response

We consider a variable selection problem for functional linear models where both multiple predictors and a response are functions. Especially we assume that variables are given as functions of time and then construct the historical functional linear model which takes the relationship of dependences of predictors and a response into consideration. Unknown parameters included in the model are estimated by the maximum penalized likelihood method with the L1 penalty. We can simultaneously estimate and select variables given as functions using the L1 type penalty. A regularization parameter involved in the regularization method is decided by a model selection criterion. The effectiveness of the proposed method is investigated by simulation studies and real data analysis.