On classes of distributions on the unit interval: structural properties and application to inequality data

TL;DR

This paper introduces two new families of distributions on the unit interval derived from gamma ratios, providing tools for modeling inequality data and estimating inequality indices like Gini and Atkinson.

Contribution

The paper presents novel distribution families on (0,1) based on gamma transformations, with explicit formulas and estimation methods, linking them to inequality measurement.

Findings

New distributions derived from gamma ratios with closed-form densities

Effective maximum likelihood estimation demonstrated via simulations

Application to real-world Gini index data shows practical relevance

Abstract

Probability distributions defined on the unit interval are widely used in fields ranging from econometrics to reliability studies. Traditional models such as the beta and Kumaraswamy distributions are well-established due to their flexibility and tractability. In this paper, we introduce two novel families of unit-interval distributions derived via non-injective transformations of the gamma ratio. These transformations, denoted $S_r$ and $T_r$, allow the construction of new random variables with support on $(0,1)$ and admit simple closed-form expressions for their densities when the underlying variables are independent gamma distributed. Notably, for $r = 1/2$, these constructions yield sample-based estimators of the Gini and Atkinson indices, establishing a direct link with classical inequality measures. We derive the distributional laws, cumulative distribution functions, quantile functions, and raw moments, and discuss maximum likelihood estimation for the proposed models. A Monte Carlo simulation study is conducted to assess the finite sample behavior of the maximum likelihood estimators under different parameter configurations. An application to cross-country Gini index data illustrates the flexibility and practical relevance of the proposed distributions in modeling real inequality indicators.