A new life of Pearson's skewness

TL;DR

This paper introduces a rigorous framework for defining and identifying true positive and negative skewness in continuous distributions using coupling and stochastic dominance methods, refining Pearson's classical skewness measure.

Contribution

It develops generalized notions of skewness called truly positive and truly negative, applying stochastic dominance to establish criteria for true skewness detection.

Findings

Defined generalized skewness using Fréchet means

Established stochastic dominance criteria for true positive skewness

Provided examples demonstrating the approach

Abstract

In this work we show how coupling and stochastic dominance methods can be successfully applied to a classical problem of rigorizing Pearson's skewness. Here, we use Fr\'{e}chet means to define generalized notions of positive and negative skewness that we call truly positive and truly negative. Then, we apply stochastic dominance approach in establishing criteria for determining whether a continuous random variable is truly positively skewed. Intuitively, this means that scaled right tail of the probability density function exhibits strict stochastic dominance over equivalently scaled left tail. Finally, we use the stochastic dominance criteria and establish some basic examples of true positive skewness, thus demonstrating how the approach works in general.