Income inequality estimation with gamma mixtures

This paper studies the estimation of the $m$th Gini index under finite mixtures of gamma distributions.

We derive closed-form expressions for the $m$th Gini index and for the expectation and bias of its non-parametric U-statistic estimator, extending previous results for both single gamma populations and gamma mixture models.

We further establish the asymptotic properties of the estimator for gamma mixtures sharing a common rate parameter, including an asymptotic lower bound for the bias, asymptotic unbiasedness, strong consistency, and asymptotic normality.

Although these theoretical results require a common rate parameter, a Monte Carlo study also investigates the estimator under mixtures with different rates and compares its performance with bias-corrected and parametric estimators.

Finally, the proposed methodology is illustrated through the analysis of an income dataset.