Estimating Income Distributions Using a Mixture of Gamma Densities

The estimation of income distributions is important for assessing income inequality and poverty and for making comparisons of inequality and poverty over time, countries and regions, as well as before and after changes in taxation and transfer policies. Distributions have been estimated both parametrically and non-parametrically. Parametric estimation is convenient because it facilitates subsequent inferences about inequality and poverty measures and lends itself to further analysis such as the combining of regional distributions into a national distribution. Non-parametric estimation makes inferences more difficult, but it does not place what are sometimes unreasonable restrictions on the nature of the distribution. By estimating a mixture of gamma distributions, in this paper we attempt to benefit from the advantages of parametric estimation without suffering the disadvantage of inflexibility. Using a sample of Canadian income data, we use Bayesian inference to estimate gamma mixtures with two and three components. We describe how to obtain a predictive density and distribution function for income and illustrate the flexibility of the mixture. Posterior densities for Lorenz curve ordinates and the Gini coefficient are obtained.