Unbiased estimation of the Gini coefficient

The Gini coefficient is a fundamental statistical measure of dispersion used widely across multiple fields. The interest in the study of the properties of the Gini coefficient is highlighted by the fact that every year the World Bank ranks the level of income inequality between countries using it. In order to calculate the coefficient, it is common practice to assume a Gamma-distributed set of values when modeling the dispersion of individual incomes in a given population. The asymptotic behavior of the sample Gini coefficient for populations following a Gamma distribution has been well-documented in the literature. However, research on the finite sample behavior has been absent due to the challenge posed by the denominator. This study aims to fill this gap by demonstrating theoretically that the sample Gini coefficient is an unbiased estimator of the population Gini coefficient for a population having Gamma ($\alpha$, $\beta$) distribution. Furthermore, our result provides a way to quantify the downward bias due to grouping when the population Gini coefficient is estimated using the sample Gini coefficient of equal-sized groups.