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.

KEY WORDS: Gini coefficient; Maximum likelihood estimation.

1. INTRODUCTION

The Lorenz curve is one of the most important tools upon which the measurement of income inequality is based. For a given economy or region, it relates the cumulative proportion of income to the cumulative proportion of population, after ordering the population according to increasing level of income. A number of approaches to Lorenz curve estimation have been adopted. In one approach, a particular assumption about the statistical distribution of income is made, the parameters of this income distribution are estimated, and a Lorenz curve consistent with the distributional assumption and consistent with the parameter estimates for that distribution is obtained. See, for example, McDonald (1984) and McDonald and Xu (1995). Ryu and Slottje (1996) suggest another approach. They approximate the Lorenz curve from any income distribution by expanding the inverse distribution function in terms of (a) an exponential polynomial series and (b) a sequence of Bernstein polynomial functions. When micro-data are available, nonparametric estimation of the Lorenz curve and related inequality measures is possible. See, for example, Beach and Davidson (1983); Gastwirth and Gail (1985); and Bishop, Chakraborti, and Thistle (1989). An alternative approach, more suited to grouped data, is to specify a particular functional form for the Lorenz curve and estimate it directly. It is this approach that is the focus of this article.