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/
and , New multi-sample nonparametric tests for panel count data. Ann. Stat. 37 (2009) 1112–1149. | DOI | MR | Zbl
and , A nonparametric test for the equality of counting processes with panel count data. Comput. Stat. Data Anal. 54 (2010) 135–142. | DOI | MR | Zbl
, and , Nonparametric comparison of several transformations of distribution functions. J. Nonparametric Stat. 25 (2013) 619–633. | DOI | MR | Zbl
R.J. Carroll, D. Ruppert, L.A. Stefanski and C. Crainiceanu, Measurement Error in Nonlinear Models: A Modern Perspective, 2nd edition. Chapman Hall, New York (2006). | MR
and , Adaptive estimation for inverse problems with noisy operators. Inverse Probl. 21 (2005) 1345–1361. | DOI | MR | Zbl
and , On inverse problems with unknown operators. IEEE Trans. Inf. Theory 47 (2001) 2876–2893. | DOI | MR | Zbl
R. Friden, Probability, Statistical Optics and Data Testing: A Problem Solving Approach. Springer Series in Information Science (2001). | Zbl
, A data-driven test to compare two or multiple time series. Comput. Stat. Data Anal. 55 (2011) 2183–2196. | DOI | MR | Zbl
and , Subsampling tests for variance changes in the presence of autoregressive parameter shifts. J. Multivariate Anal. 101 (2010) 2255–2265 | DOI | MR | Zbl
J. Meyer, Stochastic Dominance and Transformations of Random Variables, in Studies in the Economics of Uncertainty: In Honor of Josef Hadar, edited by T.B. Fomby and T.K. Seo. Springer Science and Business Media (2012) 45–58.
and , Deterministic transformations of random variables and the comparative statics of risk. J. Risk Uncertain. 2 (1989) 179–188. | DOI
and , On bootstrapping L2-type statistics in density testing. Stat. Probab. Lett. 50 (2000) 137–147 | DOI | MR | Zbl
P. Sarda and P. Vieu,Kernel regression, in Smoothing and Regression Approaches, computation and application, edited by M.G. Schimek. Wiley Series in Probability and Statistics (2000) 43–70. | Zbl
and , Estimating the Parameters of a Linear Function of a Random Variable. J. R. Stat. Soc. Ser. B 32 (1970) 362–368. | MR | Zbl
, Limiting behavior of regular functionals of empirical distributions for stationary mixing processes. Probab. Theory Relat. Fields 25 (1972) 71–82. | MR | Zbl
and , Estimation of the mean function of point processes based on data panel count data. Stat. Sin. 5 (1995) 279–290. | MR | Zbl
and , Deconvolution Estimation in Measurement Error Models: The R Package decon. J. Stat. Softw. 39 (2011) 8060.
Cité par Sources :
