Bias in estimating Theil, Atkinson, and dispersion indices for gamma mixture populations

In this paper, we derive closed-form expressions for the bias of estimators of the Theil, Atkinson, and dispersion indices when the underlying population follows a finite mixture of gamma distributions. Our methodology builds on probabilistic techniques grounded in Mosimann's proportion-sum independence theorem and the gamma-Dirichlet connection, enabling analytical tractability in the presence of population heterogeneity. These results extend existing findings for single gamma models.