Van Zwet (1964) [16] introduced the convex transformation order between two distribution functions F and G, defined by F ≤_cG if G⁻¹ ∘ F is convex. A distribution which precedes G in this order should be seen as less right-skewed than G. Consequently, if F ≤_cG, any reasonable measure of skewness should be smaller for F than for G. This property is the key property when defining any skewness measure.In the existing literature, the treatment of the convex transformation order is restricted to the class of differentiable distribution functions with positive density on the support of F. It is the aim of this work to analyze this order in more detail. We show that several of the most well known skewness measures satisfy the key property mentioned above with very weak or no assumptions on the underlying distributions. In doing so, we conversely explore what restrictions are imposed on the underlying distributions by the requirement that F precedes G in convex transformation order.

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关于范·兹威特和奥雅的偏度阶

范Zwet（1964）[16]引入了两个分布函数之间的凸变换顺序˚F和G ^，由下式定义˚F ≤ _Ç ģ如果ģ ^-1 ∘ ˚F是凸的。这之前的分配ģ以该顺序应被视为不太右偏比G.因此，如果˚F ≤ _Ç ģ，偏度的任何合理度量应该是较小˚F比ģ。该属性是定义任何偏度度量的关键属性。在现有文献中，凸变换阶的处理仅限于在F的支持下具有正密度的微分分布函数的类。这项工作的目的是更详细地分析此顺序。我们表明，一些最著名的偏度测度满足了上面提到的关键属性，而对基础分布的假设非常弱或没有假设。与此相反，我们探究了按凸变换顺序F优先于G的要求对基础分布施加了哪些限制。