Statistical Papers

, 31:209 | Cite as

Lorenz ranking of income distributions

  • A. A. Alzaid


Based on the stochastic comparison of the Lorenz curves of income distributions, five partial orderings of income distributions are obtained. Three of these orderings are the well known star shaped, stochastic and the Lorenz orderings. The other two are new and are studied in some detail. The weakest ordering which is called the Lorenz area ordering is of special importance since it enables us to compare interesting Lorenz curves. This latter ordering leads to a class of income inequality measures which are identical with the linear inequality measures considered by Mehran (1976). A discussion of these measures is presented together with an application to part of Kunzet's (1963) data.


Income Income Inequality Income Distribution Linear Measure Gini Index 
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  1. Alzaid, A. A. (1988a). Mean residual life ordering. Statist. Hefte, 29, 35–43.MathSciNetMATHGoogle Scholar
  2. Alzaid, A. A. (1988b). Length biased ordering with applications. Probab. Eng. Inf. Sc., 2, 329–341.MATHGoogle Scholar
  3. Atkinson, A. B. (1970). On the measurement of inequality. J. Econ. Theory, 2, 244–263.CrossRefGoogle Scholar
  4. Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart & Winston, New York.MATHGoogle Scholar
  5. Bawa, V. S. (1982). Stochastic dominance: A research bibliography, Management Sci. 28, 698–712.MathSciNetMATHCrossRefGoogle Scholar
  6. Blackwell, D. (1951). Comparison of experiments. Proc. Second Berkeley Symp. Math. Statist. Prob., Berkeley. University California Press, 93–102.Google Scholar
  7. Chandra, M. and Singpurwalla, N. D. (1983). Relationship between some notions which are common to reliability theory and economics. Management Sc., 113–121.Google Scholar
  8. Dagum, C. (1980). Inequality measures between income distributions with applications. Econometrica, 48, 1791–1807.CrossRefMATHGoogle Scholar
  9. DeGroot, M. H. and Eriksson, E. A. (1985). Probability forecasting, Stochastic dominance and the Lorenz curve. In Bayesian Statistics 2, Ed. Bernardo J. M. et al., Elsevier Sc. Pub. (North Holand).Google Scholar
  10. Fellman, J. (1976). The effect of transformations on Lorenz curves. Econometrica, 44, 823–824.CrossRefMathSciNetMATHGoogle Scholar
  11. Gail, M. H. and Gastwirth, J. L. (1978a). A scale-free goodness- of-fit test for exponential distribution based on the Gini Statistics. J. Roy. Statist. Soc., 40, 350–357.MathSciNetMATHGoogle Scholar
  12. Gail, M. H. and Gastwirth, J. L. (1978b). A scale-free good-ness-of-fit test for the exponential distribution based on the Lorenz curve. J. Amer. Statist. Assoc., 73, 788–793.CrossRefMathSciNetGoogle Scholar
  13. Gastwirth, J. L. (1971). A general definition of Lorenz curve. Econometrica, 39, 1037–1039.CrossRefMATHGoogle Scholar
  14. Gastwirth, J. L. (1972). The estimation of the Lorenz curve and Gini Index. Review of Economic and Statistics, 54, 306–316.CrossRefMathSciNetGoogle Scholar
  15. Hanoch, G. and H. Levy (1969). The efficiency analysis of choices involving risk, Rev. Econ. Studies, 36, 335–347.CrossRefMATHGoogle Scholar
  16. Kakwani, N. C. (1980). Income Inequality and Poverty, Method of Estimation and Policy Applications, Oxford Univ. Press.Google Scholar
  17. Karlin, S. (1968). Total Positivity, Vol. I. Stanford, Calif. Stanford University Press.MATHGoogle Scholar
  18. Keilson, J. and Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Canad. J. Statist., 10, 181–189.CrossRefMathSciNetMATHGoogle Scholar
  19. Kunzet, K. (1963). Quantative aspects of economic growth of nations: VIII Distribution of income by size, Econ. Development Cultural Change 11.Google Scholar
  20. Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Statist., 37, 1137–1153.CrossRefMathSciNetMATHGoogle Scholar
  21. Lorenz, M. O. (1905). Method of measuring the concentration of wealth. J. Amer. Statist. Assoc., 9, 209–219.Google Scholar
  22. Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications, Academic Press, New York.MATHGoogle Scholar
  23. Mehran, F. (1976). Linear measures of income inequality. Econometrica, 44, 805–809.CrossRefMathSciNetGoogle Scholar
  24. Newbery, D. M. (1970). A theorem in measurement of inequality. J. Econ. Theory, 2, 264.CrossRefMathSciNetGoogle Scholar
  25. Piesch, W. (1975). Statistische Konzentrationsmaße. J. C. B. Mohr (Paul Siebeck), Tübingen.Google Scholar
  26. Pietra, G. (1915). Dalla relazioni fra gli indici di variabilita e di concentrazioni. Atti del R. Istituto veneto di S. L. A., Anno 1914/15, Tomo LXXIV, Venezia, 1915.Google Scholar
  27. Ross, S. (1983). Stochastic Processes. Wiley, New York.MATHGoogle Scholar
  28. Rothschild, M. and Stiglitz, J. E. (1970). Increasing risk: A Definition. J. Econ. Theory, 2, 225–243.CrossRefMathSciNetGoogle Scholar
  29. Rothschild, M. and Stiglitz, J. E. (1973). Some further results on the measurement of Inequality. J. Econ. Theory, 6, 188–204.CrossRefMathSciNetGoogle Scholar
  30. Sen, A. K. (1973). On Economic Inequality, Clarendon Press, Oxford.Google Scholar
  31. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York.MATHGoogle Scholar
  32. Vinod, H. D. (1985). Measurement of economic distance between black and whites. J. Bus. Econ. Statist., 3, 78–88.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • A. A. Alzaid
    • 1
  1. 1.Department of StatisticsCollege of ScienceRiyadhSaudi Arabia

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