Identifying arbitrary transformation between the slopes in scalar-on-function regression

In this article, we study whether the slope functions of two scalar-on-function regression models in two samples are associated with any arbitrary transformation along the vertical axis. The problem is formally stated as a statistical hypothesis test, and corresponding test statistic is formed based on the estimated second derivative of the unknown transformation. The asymptotic properties of the test statistic are investigated using some advanced techniques related to the empirical process. Moreover, to implement the test for small sample size data, a bootstrap algorithm is proposed, and it is shown that the bootstrap version of the test is as good as the original test for sufficiently large sample size. Furthermore, the utility of the proposed methodology is shown for simulated datasets, and DTI data is analyzed using the proposed methodology.