  In this paper, we propose an adaptive smoothing spline (AdaSS) estimator for
the function-on-function linear regression model where each value of the
response, at any domain point, depends on the full trajectory of the predictor.
The AdaSS estimator is obtained by the optimization of an objective function
with two spatially adaptive penalties, based on initial estimates of the
partial derivatives of the regression coefficient function. This allows the
proposed estimator to adapt more easily to the true coefficient function over
regions of large curvature and not to be undersmoothed over the remaining part
of the domain. A novel evolutionary algorithm is developed ad hoc to obtain the
optimization tuning parameters. Extensive Monte Carlo simulations have been
carried out to compare the AdaSS estimator with competitors that have already
appeared in the literature before. The results show that our proposal mostly
outperforms the competitor in terms of estimation and prediction accuracy.
Lastly, those advantages are illustrated also on two real-data benchmark
examples.
