Stochastic dominance rules are well-characterized and widely used. This work aims to describe and better understand the situations when they do not hold by developing measures of stochastic non-dominance. They quantify the error caused by assuming that one random variable dominates another one when it does not. To calculate them, we search for a hypothetical random variable that satisfies the stochastic dominance relationship and is as close to the original one as possible. The Wasserstein distance between the optimal hypothetical random variable and the original one is considered as the measure of stochastic non-dominance. Depending on the conditions imposed on the probability distribution of the hypothetical random variable, we distinguish between general and specific measures of stochastic non-dominance. We derive their exact values for random variables with uniform, normal, and exponential distributions. We present relations to almost first-order stochastic dominance and to tractable almost stochastic dominance. Using monthly returns of twelve assets captured by the German stock index DAX, we solve portfolio optimization problems with the first-order and second-order stochastic dominance constraints. The measures of stochastic non-dominance allow us to compare the optimal portfolios with respect to different orders of stochastic dominance from a new angle. We also defined the closest dominating and closest approximately dominating portfolios. They brought a better understanding of the relationship between the two types of optimal portfolios. Using moving window analysis, the relationship of the in-sample measure of stochastic non-dominance to out-of-sample performance was studied, too.

中文翻译：

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投资组合优化中随机非主导性的度量


随机优势规则具有很好的特征并被广泛使用。这项工作旨在通过开发随机非优势度量来描述和更好地理解它们不成立的情况。它们量化了假设一个随机变量支配另一个随机变量而引起的误差，而实际上它并不支配。为了计算它们，我们搜索一个假设的随机变量，该变量满足随机优势关系并尽可能接近原始关系。最佳假设随机变量与原始随机变量之间的 Wasserstein 距离被认为是随机非优势的度量。根据施加在假设随机变量的概率分布上的条件，我们区分随机非优势的一般和特定度量。我们推导出具有均匀分布、正态分布和指数分布的随机变量的精确值。我们提出了与几乎一阶随机优势和可处理的几乎随机优势的关系。使用德国股票指数 DAX 捕获的 12 种资产的月回报率，我们解决了具有一阶和二阶随机优势约束的投资组合优化问题。随机指标非优势的衡量使我们能够从新的角度比较不同随机优势订单的最佳投资组合。我们还定义了最接近的主导投资组合和最接近的近似主导投资组合。他们让我们更好地理解了两种类型的最优投资组合之间的关系。使用移动窗口分析，还研究了随机非优势的样本内度量与样本外性能之间的关系。