Generalized Gini's mean difference through distortions and copulas, and related minimizing problems

TL;DR

This paper introduces generalized Gini's mean difference measures using distortions and copulas, analyzing their extremal properties within certain families of distortions to extend existing theoretical results.

Contribution

It develops new distance measures based on distortions and copulas, providing conditions for their extremal values within specific families, thus generalizing prior findings.

Findings

Conditions for existence of minima and maxima are established.

New measures extend Gini's mean difference using distortions.

Generalizations encompass recent theoretical results.

Abstract

Given a random variable $X$ and considered a family of its possible distortions, we define two new measures of distance between $X$ and each its distortion. For these distance measures, which are extensions of the Gini's mean difference, conditions are determined for the existence of a minimum, or a maximum, within specific families of distortions, generalizing some results presented in the recent literature.