  Recently, expectile-based measures of skewness akin to well-known
quantile-based skewness measures have been introduced, and it has been shown
that these measures possess quite promising properties (Eberl and Klar, 2021,
2020). However, it remained unanswered whether they preserve the convex
transformation order of van Zwet, which is sometimes seen as a basic
requirement for a measure of skewness. It is one of the aims of the present
work 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 weak expectile dispersive order
is equivalent to the dilation order. Further, we analyze the statistical
properties of empirical interexpectile ranges in some detail.
