We introduce a 2-dimensional stochastic dominance (2DSD) index to characterize both strict and almost stochastic dominance. Based on this index, we derive an estimator for the minimum violation ratio (MVR), also known as the critical parameter, of the almost stochastic ordering condition between two variables. We determine the asymptotic properties of the empirical 2DSD index and MVR for the most frequently used stochastic orders. We also provide conditions under which the bootstrap estimators of these quantities are strongly consistent. As an application, we develop consistent bootstrap testing procedures for almost stochastic dominance. The performance of the tests is checked via simulations and the analysis of real data.

翻译：我们引入一个二维随机占优（2DSD）指数来刻画严格随机占优与几乎随机占优。基于该指数，我们推导出两个变量之间几乎随机排序条件的最小违反比率（MVR，亦称临界参数）的估计量。我们确定了对于最常用随机序的经验2DSD指数与MVR的渐近性质，并给出了这些量的自助估计量具有强一致性的条件。作为应用，我们发展了适用于几乎随机占优的一致性自助检验程序。通过模拟实验及真实数据分析验证了检验方法的性能。