Mean skewness measures

TL;DR

This paper introduces a new measure of skewness that averages over all quantile-based skewness measures and provides reliable interval estimators, improving the quantification of asymmetry in distributions.

Contribution

The paper proposes a novel skewness measure that integrates over all quantile-based measures and develops interval estimators with strong coverage properties.

Findings

Interval estimators perform well across various distributions.

The new measure captures asymmetry more comprehensively.

Simulation results validate the effectiveness of the estimators.

Abstract

Skewness measures can be used to measure the level of asymmetry of a distribution. Given the prevalence of statistical methods that assume underlying symmetry, and also the desire for symmetry in order to make meaningful judgements for common summary measures (e.g. the sample mean), reliably quantifying asymmetry is an important problem. There are several measures, among them generalizations of Bowley's well known skewness coefficient, that use sample quartiles and other quantile-based measures. The main drawbacks of many measures is that they are either limited to quartiles and do not take into account more extreme tail behavior, or that they require one to choose other quantiles (i.e. choose a value for $p$ different from 0.25) in place of the quartiles. Our objective is to (i) average the skewness measures over all $p$ and (ii) provide interval estimators for the new measure with good coverage properties. Our simulation results show that the interval estimators perform very well for all distributions considered.