Semi-parametric inference on Gini indices of two semi-continuous populations under density ratio models.

This article focuses on developing semi-parametric inference methods for the Gini indices of two semi-continuous populations, where the data may include a mixture of zero values and positive skewed outcomes. The authors model each population's distribution as a mixture of a discrete mass at zero and a continuous positive component linked via a density ratio model (DRM), allowing flexible and efficient estimation without strict parametric assumptions. They propose maximum empirical likelihood estimators (MELEs) for the two Gini indices and their difference, establish their asymptotic normality, and demonstrate that these estimators are more efficient than fully nonparametric counterparts. Simulation studies and applications to real datasets—one with excessive zeros (inpatient charges by gender) and one with strictly positive values (household incomes in the Philippines)—illustrate the improved accuracy, shorter confidence intervals, and higher testing power of the proposed methods compared to existing approaches. The framework accommodates both cases with and without excessive zeros and enables inference on general functions of the two Gini indices.