F-Gintropy: An Entropic Distance Ranking Based on the Gini Index.

Tamás Sándor Biró,Máté Józsa,Zoltán Néda,András Telcs

doi:10.3390/e24030407

Abstract

We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. Its generalization to f-gintropy, using a function of the income or wealth value, distinguishes between regional inequalities more sensitively than the original construction.

Highlights

In an earlier work we introduced this quantity called gintropy [23] and demonstrated its entropy-like properties: for all income probability density function (PDF) it is non-negative, has a single maximum exactly at the average value and shows overall the corresponding convexity in terms of the cumulative rich population as well as in terms of the cumulative wealth
This article is constructed as follows: First, we review the definitions of gintropy and f-gintropy, controlling their entropy-like properties
Hungary, Romania and Japan seem to be different: an interesting and unexpected result. Such a methodology based on the gintropy instead of the commonly used probability density function could definitely be useful in cases where one prefers to group countries in clusters according to their most abundant income categories

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. It is interesting to note that the half of the Gini index can be viewed as the integral of gintropy over any of the cumulatives: G/2 = As such the G/2 value represents the area between the Lorenz curve and the diagonal in the C − F plane. To interpret the f-Gini index as an expectation of an absolute value of difference, the strict monotonity property, f ( x ) > f (y) for x > y and equivalently f ( x ) < f (y) for x < y is required Fulfilling these requirements, one obtains that G f is the half of the sum of Equations (14) and (15): h| f ( x ) − f (y)|i h| f ( x ) + f (y)|i (16).

Example

Conclusions