Invariant measures of disagreement with stochastic dominance

Abstract: An essential feature of stochastic order is its invariance against increasing maps. In this paper, we analyze a family of invariant indices of disagreement with respect to stochastic dominance. The indices in this family admit the representation $\theta(F,G)=P(X>Y)$, where $(X,Y)$ is a random vector with marginal distribution functions $F$ and $G$. This includes the case of independent marginals, but also other interesting indices related to a contamination model or to a joint quantile representation. For some choices of $\theta$ the condition $\theta(F,G)=0$ is equivalent to stochastic dominance of $G$ over $F$. We show that the index associated to the contamination model achieves the minimal value within this family. The plug-in sample-based versions of these indices lead to the Mann-Whitney, the one-sided Kolmogorov-Smirnov, and the Galton statistics. For some of the most interesting indices this fact provides sufficient theoretical support for asymptotic inference. However, this is not the case for Galton's statistic, for which we provide additional theory for its resampling behaviour. We stress on the complementary roles of some of these indices, which beyond measuring disagreement with respect to stochastic order allow to describe the maximum possible difference in status of a value $x\in \mathbb{R}$ under $F$ or $G$. We apply these indices to some real data sets.