Expectile based measures of skewness

TL;DR

This paper introduces new skewness measures based on expectiles, addressing limitations of traditional moment and quantile-based measures, and demonstrates their favorable statistical properties through theoretical analysis and simulations.

Contribution

It proposes expectile-based skewness measures that overcome drawbacks of existing methods, with derived asymptotic properties and empirical evaluation.

Findings

Expectile-based measures perform well in simulations.

They have better asymptotic behavior than moment-based measures.

They address issues of tail emphasis and discrete distribution behavior.

Abstract

In the literature, quite a few measures have been proposed for quantifying the deviation of a probability distribution from symmetry. The most popular of these skewness measures are based on the third centralized moment and on quantiles. However, there are major drawbacks in using these quantities. These include a strong emphasis on the distributional tails and a poor asymptotic behaviour for the (empirical) moment based measure as well as difficult statistical inference and strange behaviour for discrete distributions for quantile based measures. Therefore, in this paper, we introduce skewness measures based on or connected with expectiles. Since expectiles can be seen as smoothed versions of quantiles, they preserve the advantages over the moment based measure while not exhibiting most of the disadvantages of quantile based measures. We introduce corresponding empirical counterparts and derive asymptotic properties. Finally, we conduct a simulation study, comparing the newly introduced measures with established ones, and evaluating the performance of the respective estimators.