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.

翻译：随机占优因其重要局限性在实践中未获广泛应用。为提升其适用性，学界近期通过引入多种度量指标对该概念进行拓展，这些指标用于量化一个概率分布对另一个分布的随机占优程度。本文从基本原理出发，借助简明示例，在要求保持递增函数不变性的前提下，系统阐述并探讨了部分此类指标。这自然导向具有吸引力的通用表达形式 $\theta(F,G)=P(X>Y)$，其中 $(X, Y)$ 是边缘分布为 $F$ 和 $G$ 的随机向量。本文考察的指标源于 $X$ 与 $Y$ 间的不同依赖结构，包括边缘独立的情形，以及与污染模型或联合分位数表示相关的其他指标。我们着重强调部分指标的互补作用：它们不仅能度量对随机占优的偏离程度，还能描述值 $x\in \Rea$ 在分布 $F$ 或 $G$ 下状态的最大可能差异。我们将这些指标应用于模拟及真实数据集，探究其实际优势与局限。研究涵盖了 Mann-Whitney、单尾 Kolmogorov-Smirnov 和 Galton 秩统计量等经典统计量中较少被关注的特性，甚至为后者提供了新的理论补充。