Gini-stable Lorenz curves and their relation to the generalised Pareto distribution

TL;DR

This paper introduces a new family of Lorenz curves that are Gini-stable and relate to the Generalised Pareto Distribution, providing insights into inequality measurement and distribution modeling.

Contribution

It establishes a novel iterative process for generating Lorenz curves with fixed Gini index and links these to a unified parametrization of the Generalised Pareto Distribution.

Findings

Lorenz curves are stochastically ordered by sample size and Gini index.

Asymptotic Lorenz curves correspond to a new parametrization of the GPD.

The model fits socioeconomic and environmental data well.

Abstract

We introduce an iterative discrete information production process where we can extend ordered normalised vectors by new elements based on a simple affine transformation, while preserving the predefined level of inequality, G, as measured by the Gini index. Then, we derive the family of empirical Lorenz curves of the corresponding vectors and prove that it is stochastically ordered with respect to both the sample size and G which plays the role of the uncertainty parameter. We prove that asymptotically, we obtain all, and only, Lorenz curves generated by a new, intuitive parametrisation of the finite-mean Pickands' Generalised Pareto Distribution (GPD) that unifies three other families, namely: the Pareto Type II, exponential, and scaled beta distributions. The family is not only totally ordered with respect to the parameter G, but also, thanks to our derivations, has a nice underlying interpretation. Our result may thus shed a new light on the genesis of this family of distributions. Our model fits bibliometric, informetric, socioeconomic, and environmental data reasonably well. It is quite user-friendly for it only depends on the sample size and its Gini index.