Robust statistical comparison of random variables with locally varying scale of measurement
Christoph Jansen, Georg Schollmeyer, Hannah Blocher, Julian Rodemann, Thomas Augustin
Abstract
Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.
BibTeX
@InProceedings{pmlr-v216-jansen23a,
title = {Robust statistical comparison of random variables with locally varying scale of measurement},
author = {Jansen, Christoph and Schollmeyer, Georg and Blocher, Hannah and Rodemann, Julian and Augustin, Thomas},
booktitle = {Proceedings of the Thirty-Ninth Conference on Uncertainty in Artificial Intelligence},
pages = {941--952},
year = {2023},
editor = {Evans, Robin J. and Shpitser, Ilya},
volume = {216},
series = {Proceedings of Machine Learning Research},
month = {31 Jul--04 Aug},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v216/jansen23a/jansen23a.pdf},
url = {https://proceedings.mlr.press/v216/jansen23a.html},
abstract = {Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.}
}