On the Skewness Order of van Zwet and Oja

Abstract

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.