In this work we show how coupling and stochastic dominance methods can be successfully applied to a classical problem of rigorizing Pearson's skewness. Here, we use Fr\'{e}chet means to define generalized notions of positive and negative skewness that we call truly positive and truly negative. Then, we apply stochastic dominance approach in establishing criteria for determining whether a continuous random variable is truly positively skewed. Intuitively, this means that scaled right tail of the probability density function exhibits strict stochastic dominance over equivalently scaled left tail. Finally, we use the stochastic dominance criteria and establish some basic examples of true positive skewness, thus demonstrating how the approach works in general.
翻译:在这项工作中,我们展示了如何成功地将混合和随机主导法运用到对皮尔逊的扭曲性的传统问题中。 在这里,我们使用 Fr\'{e}chet 来定义我们称之为真正正反的正和负的正和负扭曲性的普遍概念。 然后,我们运用随机主导法来制定标准,以确定连续随机变量是否真正正倾斜。 直观地说,这意味着概率密度函数的右尾巴的伸缩显示对等的左尾巴的严格扭曲主导权。 最后,我们使用随机主导法标准,建立一些真正的正对立法的基本例子,从而展示该方法在总体上是如何运作的。