Robust Statistical Comparison of Random Variables with Locally Varying Scale of Measurement
This work addresses a foundational issue in statistics and machine learning for comparing variables in non-standard spaces, though it appears incremental as it builds on existing orders like stochastic dominance.
The authors tackled the problem of comparing random variables in spaces with locally varying measurement scales by proposing a generalized stochastic dominance order that includes stochastic dominance and expectation order as extremes. They developed a regularized statistical test for this order, operationalized via linear optimization and robustified with imprecise probability models, demonstrating applications in poverty measurement, finance, and medicine.
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.