Dette, HolgerTang, Jiajun2021-10-152021-10-152021http://hdl.handle.net/2003/40526http://dx.doi.org/10.17877/DE290R-22397We propose a reproducing kernel Hilbert space approach to estimate the slope in a function-on-function linear regression via penalised least squares, regularized by the thin-plate spline smoothness penalty. In contrast to most of the work on functional linear regression, our main focus is on statistical inference with respect to the sup-norm. This point of view is motivated by the fact that slope (surfaces) with rather different shapes may still be identified as similar when the difference is measured by an L2-type norm. However, in applications it is often desirable to use metrics reflecting the visualization of the objects in the statistical analysis. We prove the weak convergence of the slope surface estimator as a process in the space of all continuous functions. This allows us the construction of simultaneous confidence regions for the slope surface and simultaneous prediction bands. As a further consequence, we derive new tests for the hypothesis that the maximum deviation between the “true” slope surface and a given surface is less or equal than 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. As a by-product, we also derive several new results and statistical inference tools for the function-on-function linear regression model, such as minimax optimal convergence rates and likelihood-ratio tests. 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.enfunction-on-function linear regressionmaximum deviationreproducing kernel Hilbert spacebootstraprelvant hypothesessimultaneous confidence regionsminimax optimality310330620Text