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

Roberto Vila, Helton Saulo, Poliana Matos, Subhankar Dutta

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.

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

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 and , allow the construction of new random variables with support on and admit simple closed-form expressions for their densities when the underlying variables are independent gamma distributed. Notably, for , 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.

Paper Structure

This paper contains 30 sections, 5 theorems, 90 equations, 40 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The unit probabilistic models
  3. Deriving the laws of $W$ and $Z$
  4. Distribution and density functions of $W$ and $Z$
  5. Quantiles
  6. Closed-form expressions for the $n$th raw moments of $W$ and $Z$
  7. The $1$st raw moment of $W$: case $r=1/2$ and $1/3<\lambda_2/\lambda_1<2$
  8. The $n$th raw moment of $W$: case $r\in\mathbb{N}$ and $\lambda_1=\lambda_2$
  9. The $1$st raw moment of $Z$: case $r=1/2$ and $\lambda_2/\lambda_1<2$
  10. The $n$th raw moment of $Z$: case $r\in\mathbb{N}$ and $\lambda_1=\lambda_2$
  11. Maximum likelihood estimation
  12. Simulation Results
  13. Monte Carlo Simulation
  14. Results for model $W$
  15. Results for model $Z$
  16. ...and 15 more sections

Key Result

Proposition A.1

If $X\sim\text{Gamma}(\alpha_1,\lambda_1)$ and $Y\sim\text{Gamma}(\alpha_2,\lambda_2)$, then, we have: and

Figures (40)

  • Figure 1: Plots of the PDFs $f_W$ (left) and $f_Z$ (right) with varying parameters $\alpha_1, \alpha_2, \lambda_1$ and $\lambda_2$, with $r=0.4$.
  • Figure 2: Plots of the PDFs $f_W$ (left) and $f_Z$ (right) with varying parameters $\alpha_1, \alpha_2, \lambda_1$ and $\lambda_2$, with $r=1$.
  • Figure 3: Plots of the PDFs $f_W$ (left) and $f_Z$ (right) with varying parameters $\alpha_1, \alpha_2, \lambda_1$ and $\lambda_2$, with $r=2$.
  • Figure 4: Bias and RMSE of the estimators under model $W$ for $(\alpha_1,\alpha_2)=(0.5,0.5)$.
  • Figure 5: Bias and RMSE of the estimators under model $W$ for $(\alpha_1,\alpha_2)=(0.8,1.0)$.
  • ...and 35 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Remark 4.1
  • Remark 4.2
  • Remark 6.1
  • Remark 6.2
  • Remark 6.3
  • Proposition A.1
  • Proposition A.2
  • proof
  • ...and 5 more