However, according to this relaxation, X ∼ F is indeed stochastically larger than Y ∼ G. Almost Stochastic Dominance To overcome the limitations of the above straight forward approach, and define a relaxation of stochastic order, we turn to a definition that is based on the proportion of points in the domain of the participating distributions for which SO holds.
the test we will introduce below is based on the following two violation sets: Intuitively, the variable with the smaller violation set should be declared superior and the ratio between these sets should define the gap between the distributions.