Robust Statistical Comparison of Random Variables with Locally Varying Scale of Measurement

Christoph Jansen, Georg Schollmeyer, Hannah Blocher, Julian Martin Rodemann, Thomas Augustin

Keywords: generalized stochastic dominance, non-standard scale of measurement, robust testing, regularized testing, credal sets, imprecise probabilities, multivariate data with differently scaled dimensions

TL;DR: We consider a generalization of stochastic dominance to handle data with non-standard scale of measurement and introduce corresponding (regularized and/or robustified) statistical tests.

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.