This article studies the problem whether two convex (concave) regression functions modelling the relation between a response and covariate in two samples differ by a shift in the horizontal and/or vertical axis. We consider a nonparametric situation assuming only smoothness of the regression functions. A graphical tool based on the derivatives of the regression functions and their inverses is proposed to answer this question and studied in several examples. We also formalize this question in a corresponding hypothesis and develop a statistical test. The asymptotic properties of the corresponding test statistic are investigated under the null hypothesis and local alternatives. In contrast to most of the literature on comparing shape invariant models, which requires independent data the procedure is applicable for dependent and non-stationary data. We also illustrate the finite sample properties of the new test by means of a small simulation study and two real data examples.

中文翻译：

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识别两条回归曲线之间的转变

本文研究对两个样本中的响应和协变量之间的关系进行建模的两个凸（凹）回归函数是否因水平轴和/或垂直轴的偏移而不同。我们考虑非参数情况，仅假设回归函数是平滑的。提出了一种基于回归函数的导数及其逆函数的图形工具来回答这个问题，并在几个例子中进行了研究。我们还在相应的假设中正式化了这个问题，并开发了一个统计检验。在零假设和局部替代方案下研究相应检验统计量的渐近特性。与大多数关于比较形状不变模型的文献相反，它需要独立数据，该过程适用于相关数据和非平稳数据。