Nonparametric comparison of several transformations of distribution functions

This paper considers two random variables such that there exists a monotone transformation between their distribution functions. The problem is to test if there is a change in this transformation when these two variables are observed under K different conditions. The approach considered is a CUSUM test based on the cumulative sum of the residuals and a test statistic is proposed for testing the equality of the K transformations. The asymptotic distribution of the test statistic is derived and its finite sample properties are examined by simulation. As a further illustration, an analysis of a real data set concerning the impact of the financial crisis of September 2008 is given.