Theoretical and empirical Lorenz functions, Gini indices, and their properties

The issues of assessing the fairness and efficiency of the distribution of the total income of society between different groups of the population have attracted attention of scientists for a long time. They became most relevant at the end of the 19th – beginning of the 20th centuries in connection with the intensive stratification of countries with various political and social systems caused by the intensive development of the economy, science and technology. The Lorenz function and the Lorenz curve, as well as the Gini index, are commonly used for theoretical research and applications in the economic and social sciences. These tools were originally introduced to describe and study the inequality in the incomes and wealth distribution among a given population. Nowadays they have found wide application in such fields as demography, insurance, healthcare, the risk and reliability theory, as well as in other areas of human activities. In this paper we present the properties of the Lorentz function and various representations of the Gini index, systematize the analytical results for uniform, exponential, power-law (types I and II) and lognormal distributions, as well as for the Pareto distribution (types I and II). Additionally, the issue of estimating inequality based on the Pietra index and its relationship with the Lorentz function was studied. Nonparametric estimates of the Lorentz function and the Gini index based on a sample from the corresponding distribution are considered. Strict consistency and asymptotic unbiasedness of these estimates are shown under certain conditions for the initial distribution with an increase in the sample size. On the basis of the method of linearization of estimates, the asymptotic normality of the empirical Lorentz function and the empirical Gini index is determined.