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.
Accepté le :
DOI : 10.1051/ps/2016003
Mots-clés : Empirical cumulative distribution, nonlinear transformation, nonparametric estimation
@article{PS_2016__20__510_0,
author = {Boutahar, Mohamed and Pommeret, Denys},
title = {A test for the equality of monotone transformations of two random variables},
journal = {ESAIM: Probability and Statistics},
pages = {510--526},
publisher = {EDP-Sciences},
volume = {20},
year = {2016},
doi = {10.1051/ps/2016003},
mrnumber = {3581832},
zbl = {1357.62191},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2016003/}
}
TY - JOUR AU - Boutahar, Mohamed AU - Pommeret, Denys TI - A test for the equality of monotone transformations of two random variables JO - ESAIM: Probability and Statistics PY - 2016 SP - 510 EP - 526 VL - 20 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2016003/ DO - 10.1051/ps/2016003 LA - en ID - PS_2016__20__510_0 ER -
%0 Journal Article %A Boutahar, Mohamed %A Pommeret, Denys %T A test for the equality of monotone transformations of two random variables %J ESAIM: Probability and Statistics %D 2016 %P 510-526 %V 20 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2016003/ %R 10.1051/ps/2016003 %G en %F PS_2016__20__510_0
Boutahar, Mohamed; Pommeret, Denys. A test for the equality of monotone transformations of two random variables. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 510-526. doi : 10.1051/ps/2016003. http://www.numdam.org/articles/10.1051/ps/2016003/
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