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arXiv · 2508.01154

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

Roberto Vila·Helton Saulo·Poliana Matos·Subhankar Dutta
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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.

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Roberto Vila, Helton Saulo, Poliana Matos, Subhankar Dutta. 2025-08-02. On classes of distributions on the unit interval: structural properties and application to inequality data. https://arxiv.org/abs/2508.01154

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