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 whether these measures preserve the convex transformation order of van Zwet, which is a basic requirement for a measure of skewness. It is one aim of this paper to answer this question in the affirmative. These measures of skewness are scaled using interexpectile distances. We introduce orders of variability based on these quantities and show that the so-called expectile dispersive order is equivalent to the dilation order. Further, we analyze the statistical properties of empirical interexpectile ranges in some detail.