Income Modeling and Balancing pp 29-54 | Cite as

# Lorenz Densities and Lorenz Curves

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## Abstract

Lorenz curves and Lorenz densities are introduced for real-valued random variables with finite and strictly positive expectation. Gastritic’s definition of a Lorenz curve is used. Basic properties of Lorenz curves are given as well as approximation results. Interestingly, when a sequence of distribution functions converges to a limit distribution function, the corresponding sequence of Lorenz curves need not converge to the Lorenz curve of the limit distribution function. Yet, convergence can be ensured under sufficient conditions. The characterizations of the function sets that are equal to either all Lorenz curves or all Lorenz densities are stated both. Examples of Lorenz curves are given including the Lorenz curve of the Cantor distribution. Some principles to derive Lorenz curves from other Lorenz curves are shown and finally, inequality measures based on Lorenz curves are given, with the Gini index being the most prominent example.

## Keywords

Generalize Inverse Gini Index Lorenz Curve Support Point Empirical Distribution Function## References

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