A test for the equality of monotone transformations of two random variables

Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France.
denys.pommeret@univ-amu.fr

Received: 19 March 2015
Revised: 4 January 2016
Accepted: 2 February 2016

Abstract

It is frequent that observations arise from a random variable modified by an unknown transformation. This problem is considered in a two-sample context when two random variables are perturbed by two unknown transformations. We propose a test for the equality of those transformations. Two cases are considered: first, the two random variables have known distributions. Second, they have unknown distributions but they are observed before transformations. We propose nonparametric test statistics based on empirical cumulative distribution functions. In the first case the asymptotic distribution of the test statistic is the standard normal distribution. In the second case it is shown that the asymptotic distribution is a convolution of exponential distributions. The convergence under contiguous alternatives is studied. Monte Carlo studies are performed to analyze the level and the power of the test. An illustration is presented through a real data set.