Estimating Lorenz curves using a Dirichlet distribution

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified, it is typically estimated by linear or nonlinear least squares estimation techniques that have good properties when the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This article proposes and applies a new methodology that recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution, Five Lorenz curve specifications are used to demonstrate the technique. Maximum likelihood estimates under the Dirichlet distribution assumption provide better fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.