Statistical inference for function-on-function linear regression

We propose a reproducing kernel Hilbert space approach for statistical inference regarding the slope in a functionon-function linear regression via penalised least squares, regularized by the thin-plate spline smoothness penalty. We derive a Bahadur expansion for the slope surface estimator and prove its weak convergence as a process in the space of all continuous functions. As a consequence of these results, we construct minimax optimal estimates, simultaneous confidence regions for the slope surface and simultaneous prediction bands. Moreover, we derive new tests for the hypothesis that the maximum deviation between the "true" slope surface and a given surface is less than or equal to a given threshold. In other words, we are not trying to test for exact equality (because in many applications this hypothesis is hard to justify), but rather for pre-specified deviations under the null hypothesis. To ensure practicability, non-standard bootstrap procedures are developed addressing particular features that arise in these testing problems. We also demonstrate that the new methods have good finite sample properties by means of a simulation study and illustrate their practicability by analyzing a data example.