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Lorenz Densities and Lorenz Curves

  • Thomas Kämpke
  • Franz Josef Radermacher
Chapter
  • 664 Downloads
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 679)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Arnold BC (2008) The Lorenz curve: evergreen after 100 years. In: Advances on income inequality and concentration measures. Routledge, London, pp 12–24Google Scholar
  2. Bauer H (1974) Wahrscheinlichkeitstheorie und Grundzüge der Maßtheorie, 2nd edn. deGruyter, BerlinGoogle Scholar
  3. Chotikapanik D (ed) (2008) Modeling income distributions and Lorenz curves. Springer, New YorkGoogle Scholar
  4. De Maio FG (2007) Income inequality measures. J Epidemiol Community Health 61:849–852CrossRefGoogle Scholar
  5. Donoghue WF (1969) Distributions and Fourier transforms. Academic, New YorkGoogle Scholar
  6. Elstrodt J (2009) Maß- und Integrationstheorie, 6th edn. Springer, BerlinGoogle Scholar
  7. Farris FA (2010) The Gini index and measures of inequality. Am Math Mon 117:851–864CrossRefGoogle Scholar
  8. Friedman B (1940) A note on convex functions. Bull Am Math Soc 46:473–474CrossRefGoogle Scholar
  9. Gastwirth JL (1971) A general definition of the Lorenz curve. Econometrica 39:1037–1039CrossRefGoogle Scholar
  10. Goldie CM (1977) Convergence theorems for empirical Lorenz curves and their inverses. Adv Appl Probab 9:765–791CrossRefGoogle Scholar
  11. Iritani J, Kuga K (1983) Duality between the Lorenz curves and the income distribution functions. Econ Stud Q 34:9–21Google Scholar
  12. Leslie RA (2000) Exploring the Gini Index of inequality with Derive. Working document, Agnes Scott College, DecaturGoogle Scholar
  13. Rockafeller RT (1972) Convex analysis, 2nd edn. Princeton University Press, PrincetonGoogle Scholar
  14. Royden HL (1968) Real analysis, 2nd edn. Macmillan, LondonGoogle Scholar
  15. Sarabia JM, Jordá V (2013) Modeling bivariate Lorenz curves with applications to multidimensional inequality. In: Fifth meeting of the Society for the Study of Economic Inequality, Bari, 38 ppGoogle Scholar
  16. Thistle PD (1989) Duality between generalized Lorenz curves and distribution functions. Econ Stud Q 40:183–187Google Scholar
  17. Thompson WA (1976) Fisherman’s luck. Biometrics 32:265–271CrossRefGoogle Scholar
  18. Yitzhaki S, Olkin I (1991) Concentration indices and concentration curves. In: Stochastic orders and decision under risk, pp 380–392Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Thomas Kämpke
    • 1
  • Franz Josef Radermacher
    • 2
  1. 1.Research Institute for Applied Knowledge Processing (FAW/n)UlmGermany
  2. 2.Department of Computer ScienceUniversity of UlmUlmGermany

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