The Beta–Pareto Distribution and Its Impact on the Socio–Economic Environment

Abstract

The Beta–Pareto distribution is a flexible model for heavy–tailed data of the kind
often seen in socio–economic studies, finance, insurance, and reliability. In this paper
we study the distribution from both a theoretical and an inferential point of view.
We summarize its main properties, including a convenient stochastic representation,
the quantile function, moments, tail behavior, hazard rate features, and Shannon
entropy. We also derive explicit expressions for common inequality tools, notably
the Lorenz curve and the Gini coefficient.
For inference, we develop maximum likelihood estimation for both complete sam-
ples and right–censored data, and we discuss the large–sample behavior of the result-
ing estimators. We then compare two standard optimization strategies—Newton–
Raphson and conjugate gradient—focusing on convergence and numerical stability.
A Monte Carlo study explores how the estimators behave in heavy–tailed settings
that resemble income data.
Finally, we apply the model to income and insurance datasets. Using information
criteria and goodness–of–fit diagnostics, we show that the Beta–Pareto distribution
captures upper–tail concentration and inequality patterns more effectively than sev-
eral classical Pareto–type alternatives. Overall, the results suggest that the Beta–
Pareto model is both interpretable and practically useful, with conjugate gradient
methods providing reliable estimation in our experiments.