Extensions of true skewness for unimodal distributions

TL;DR

This paper extends the concept of true skewness for continuous distributions, providing new criteria, establishing true skewness for several distributions, and exploring extensions to discrete and multivariate cases.

Contribution

It introduces novel criteria for true skewness, applies them to multiple distributions, and discusses extensions to discrete and multivariate settings.

Findings

Established true skewness for Weibull, Lévy, skew-normal, and chi-squared distributions.

Proposed new criteria for true skewness.

Discussed extension of true skewness to discrete and multivariate distributions.

Abstract

A 2022 paper arXiv:2009.10305v4 introduced the notion of true positive and negative skewness for continuous random variables via Fr\'echet $p$-means. In this work, we find novel criteria for true skewness, establish true skewness for the Weibull, L\'evy, skew-normal, and chi-squared distributions, and discuss the extension of true skewness to discrete and multivariate settings. Furthermore, some relevant properties of the $p$-means of random variables are established.