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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 in two real-data benchmark examples. The AdaSS estimator is implemented in the  package , openly available online on CRAN.<\/jats:p>","DOI":"10.1007\/s00180-022-01223-6","type":"journal-article","created":{"date-parts":[[2022,4,29]],"date-time":"2022-04-29T12:16:57Z","timestamp":1651234617000},"page":"191-216","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Adaptive smoothing spline estimator for the function-on-function linear regression model"],"prefix":"10.1007","volume":"38","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5007-525X","authenticated-orcid":false,"given":"Fabio","family":"Centofanti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Lepore","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessandra","family":"Menafoglio","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Biagio","family":"Palumbo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Simone","family":"Vantini","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,4,29]]},"reference":[{"issue":"3","key":"1223_CR1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/0378-3758(95)00021-6","volume":"49","author":"F Abramovich","year":"1996","unstructured":"Abramovich F, Steinberg DM (1996) Improved inference in nonparametric regression using lk-smoothing splines. 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The Swedish mortality dataset is available from the Human Mortality Database (). For confidentiality reasons, the ship CO<sub>2<\/sub> emission dataset is available upon request.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Data Availability Statement"}},{"value":"The authors declare that they have no conflict of interest.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}