AI 中文总结
该研究在波兰空间上提出带参数的几乎随机优势参数类，通过最优传输确定最优γ，证明其传递性，涵盖Müller等的多元方法并验证鲁棒性。
AI 中文摘要
我们在一般波兰空间（Polish spaces）上研究几乎随机优势的参数类，将其作为带参数γ∈[0,1]的概率分布序关系。γ值越大对应序关系越弱：γ=0对应经典随机优势≤_st，γ=1对应基于固定增函数g的期望比较的完全预序。已知X≤_st可通过最优传输问题的解表征，即存在合适代价函数c使得OT_c(X,Y)=0。我们推广该思路，可通过最优传输问题的解确定几乎随机优势的最优γ参数。利用经典Kantorovich-Rubinstein对偶定理到拟伪度量的推广，我们基于参数类测试函数的期望比较推导该序的对偶表征，因此我们的关系始终是传递的，这与近期一些基于最优传输的几乎随机优势方法不同。Müller等人（2025）近期提出了基于偏导数有界的测试函数类的几乎随机优势多元方法，我们证明该方法是我们框架的特例，并推导了该文献中实例及其他实例的最优γ参数。我们还证明了鲁棒性结果：在与最优传输问题相关的Wasserstein型度量下，分布受小扰动时，最优γ仅小幅增大。
英文摘要
We study parametric classes of almost stochastic dominance on general Polish spaces as order relations for probability distributions with a parameter $γ\in [0,1]$. Larger values of $γ$ correspond to weaker order relations: $γ=0$ gives classical stochastic dominance $\le_{st}$, whereas $γ=1$ gives a complete preorder based on comparison of expectations of a fixed increasing function $g$. It is well known that $X \le_{st} Y$ can be characterized by the existence of a solution to an optimal transport problem with $\mathrm{OT}_c(X,Y)=0$ for a suitable cost function $c$. We generalize this idea so that the best possible parameter $γ$ for almost stochastic dominance can be determined from the solution of an optimal transport problem. Using a generalization of the classical Kantorovich--Rubinstein duality theorem to quasi-pseudo-metrics, we derive a dual characterization of the order in terms of expectation comparisons for a parametric class of test functions. Consequently, our relations are always transitive, in contrast to some other recent approaches to almost stochastic dominance based on optimal transport. A natural multivariate approach to almost stochastic dominance, based on classes of test functions with bounds on partial derivatives, was recently introduced by Müller et al. (2025). We show that this approach is a special case of our framework and derive the best possible parameters $γ$ for examples considered there, as well as for other examples from the literature. We also prove a robustness result showing that, under small perturbations of the distributions in a Wasserstein-type metric related to the optimal transport problem, the best possible $γ$ increases only slightly.
