Invariant measures of disagreement with stochastic dominance

TL;DR

This paper explores invariant measures of disagreement based on stochastic dominance, introducing indices that quantify the degree of dominance and analyzing their properties through examples, applications, and theoretical insights.

Contribution

It introduces and discusses new indices for measuring disagreement under stochastic dominance that are invariant under increasing transformations and explores their theoretical and practical implications.

Findings

Indices can measure the degree of stochastic dominance disagreement.

Some indices reveal the maximum difference in distributional status.

Applications demonstrate practical advantages and limitations.

Abstract

Stochastic dominance has not been too employed in practice due to its important limitations. To increase its versatility, the concept has recently been adapted by introducing various indices that measure the degree to which one probability distribution stochastically dominates another. In this paper, starting from the fundamentals and using very simple examples, we present and discuss some of these indices when one intends to maintain invariance through increasing functions. This naturally leads to consideration of the appealing common representation, $\theta(F,G)=P(X>Y)$, where $(X, Y)$ is a random vector with marginal distributions $F$ and $G$. The indices considered here arise from different dependencies between X and Y. This includes the case of independent marginals, as well as other indices related to a contamination model or to a joint quantile representation. We emphasize the complementary role of some of these indices, which, in addition to measuring disagreement with respect to stochastic dominance, enable us to describe the maximum possible difference in the status of a value $x\in \Rea$ under $F$ or $G$. We apply these indices to simulated and real-world datasets, exploring their practical advantages and limitations. The tour includes lesser-known facets of well-known statistics such as Mann-Whitney, one-tailed Kolmogorov-Smirnov and Galton's rank statistics, even providing additional theory for the latter.