An Analysis of Some Properties of Alternative Measures of Income Inequality Based on the Gamma Distribution Function

Abstract

The Gini, Theil entropy, and Pietra measures of inequality associated with the gamma distribution function are expressed in terms of the parameters defining the gamma distribution. Method of moments (MME) and maximum likelihood estimators (MLE) of these measures are obtained along with expressions for the asymptotic standard errors of the MLE measures. A table is presented that facilitates the calculation of MLE of inequality measures and the associated asymptotic standard errors by expressing each as a function of the ratio of the arithmetic mean and geometric mean. This table also facilitates the calculation of MME estimates of inequality measures. The results of a Monte Carlo study are used to compare the performance of the MME and MLE for data generated from a population characterized by a gamma distribution and to consider questions of statistical inference and requisite sample size.