Functional linear regression for functional response via sparse basis selection

We study a sparse estimation in functional linear regression model for functional response where the bivariate regression coefficient function takes zero values in a certain region of domain, so it is generated by a sparse set of basis functions. From a variable selection perspective, we construct a sparse basis representation for the coefficient function using the penalized least squares method. The proposed method enables us to simultaneously estimate the regression parameters and select basis functions. For a given basis, we show that our approach consistently identifies true subset of basis functions and the resulting estimator has asymptotically the same properties as the oracle estimator derived from the true underlying model. Simulation studies and a real data application are provided to demonstrate a finite sample performance of the proposed method.