Stochastic orders and measures of skewness and dispersion based on expectiles

Recently, expectile-based measures of skewness have been introduced which possess quite promising properties (Eberl and Klar, 2021, 2020). However, it remained unanswered if these measures preserve the convex transformation order of van Zwet, which is a basic requirement for a measure of skewness. These measures are scaled using interexpectile distances. Here, it is not clear if these measures of variability preserve the dispersive ordering. It is the main aim of this paper to answer both questions in the affirmative. Moreover, we study the interexpectile range in some detail.