Some indices to measure departures from stochastic order

We consider indices, $\theta(F,G)$, to measure the disagreement with the stochastic dominance of the distribution function $G$ over another one $F$, satisfying that there exist random variables $X,Y$ with these respective distribution functions, such that $\theta(F,G)=P(X>Y)$. We analyze the indices corresponding to an independent (the $\rho$-index), a contamination (the $\pi$-index) and a quantile (the $\gamma$-index) frameworks; the cases $\pi(F,G)=0$ and $\gamma(F,G)=0$ being equivalent to the stochastic dominance of $G$ over $F$. Notably, we show that $\pi(F,G)$ coincides with the minimum possible value of $P(X>Y)$. The plug-in sample-based versions of these indices lead to the Mann-Whitney, the one-sided Kolmogorov-Smirnov, and the Galton statistics. Asymptotic theories are available which are sufficient for $\rho$ and $\pi$. Regrettably, these theories do not cover the bootstrap approximation for Galton's statistic. Then, we provide some additional theory for its resampling behaviour. Moreover, our discussion leads to the consideration of $\gamma$ and $\pi$ as complementary, with $\gamma$ describing the level of disagreement with the stochastic dominance while $\pi$ measures 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.