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

This paper examines the finite-sample bias of estimators for the Theil and Atkinson indices, as well as for the variance-to-mean ratio (VMR), under the assumption that the population follows a finite mixture of gamma distributions with a common rate parameter. Using Mosimann's proportion-sum independence theorem and the structural relationship between the gamma and Dirichlet distributions, these estimators were rewritten as functions of Dirichlet vectors, which enabled the derivation of closed-form analytical expressions for their expected values. A Monte Carlo simulation study evaluates the performance of both the traditional and bias-corrected estimators across a range of mixture scenarios and sample sizes, revealing systematic bias induced by population heterogeneity and demonstrating the effectiveness of the proposed corrections, particularly in small and moderate samples. An empirical application to global per capita GDP data further illustrates the practical relevance of the methodology and confirms the suitability of gamma mixtures for representing structural economic heterogeneity.