In this article, we study whether the slope functions of two functional regression models in two samples are associated with any arbitrary transformation (barring constant and linear transformation) or not along the vertical axis. In order to address this issue, a statistical testing of the hypothesis problem is formalized, and the test statistic is formed based on the estimated second derivative of the unknown transformation. The asymptotic properties of the test statistics 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 data sets, and DTI data is analyzed using this methodology.

翻译：本文研究两个样本中函数回归模型的斜率函数是否沿纵轴存在任意变换（常数变换和线性变换除外）。为解决此问题，我们形式化地提出了假设检验的统计框架，并基于未知变换的估计二阶导数构建检验统计量。利用与经验过程相关的高级技术，我们研究了该检验统计量的渐近性质。此外，为在小样本数据中实施检验，本文提出了一种Bootstrap算法，并证明当样本量足够大时，该算法的Bootstrap版本检验效果与原检验相当。最后，通过模拟数据集展示了所提方法的实用性，并运用该方法分析了DTI数据。