The Gini coefficient and discontinuity - Jens Peter Kristensen (Cogent Economics & Finance, 2022)
This article reveals a discontinuity in the mapping from a Lorenz curve to the associated cumulative distribution function. The problem is of a mathematical nature-based on an analysis of the transformation between the distribution function of a bound random variable and its Lorenz curve. It will be proven that the transformation from a normalized income distribution to its Lorenz curve is a continuous bijection with respect to the Lq ([0,1])-metric-for every q Ï 1. The inverse transformation, however, is not continuous for any q Ï 1. This implies a more careful attitude when interpreting the value of a Gini coefficient. A further problem is that if you have estimated a Lorenz curve from empirical data,then you cannot trust that the associated distribution is a good estimate of the true income distribution.
Keywords: 330; inequality; probability; math analysis; discontinuity