Some Generalized Functions for the Size Distribution of Income

  • James B. McDonald
Part of the Economic Studies in Equality, Social Exclusion and Well-Being book series (EIAP, volume 5)


Many distributions have been used as descriptive models for the size distribution of income. This paper considers two generalized beta distributions which include many of these models as special or limiting cases. These generalized distributions have not been used as models for the distribution of income and provide a unified method of comparing many models previously considered.

Expressions are reported which facilitate parameter estimation and the analysis of associated means, variances, and various measures of inequality.

The distributions considered are fit to US family income and their relative performance is compared.


Income Distribution Pareto Distribution Generalize Gamma Generalize Beta Current Population Report 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aitchison, J. and J. A. C. Brown (1969) The Lognormal Distribution with Special References to its Uses in Economics, Cambridge University Press, Cambridge. Google Scholar
  2. Arnold, B. (1980) Pareto Distributions: Pareto and Related Heavy-tailed Distributions, mimeographed manuscript, University of California at Riverside.Google Scholar
  3. Atoda, N., T. Suruga and T. Tachibanaki (1980) Statistical Inference of Functional Forms for Income Distribution, manuscript. Kyoto University.Google Scholar
  4. Cox, D. R. and D. V. Hinckley (1974) Theoretical Statistics, Chapman and Hall, London.Google Scholar
  5. Cronin, D. C. (1979) A Function for the Size Distribution of Income: A Further Comment, Econometrica, 47, 773-774.CrossRefGoogle Scholar
  6. Esteban, J. (1981) Income-Share Elasticity, Density Functions and the Size Distribution of Income, mimeographed manuscript, University of Barcelona. Google Scholar
  7. Gastwirth, J. L. (1972) The Estimation of the Lorenz Curve and Gini Index, Review of Economics and Statistics, 54, 306-316.CrossRefGoogle Scholar
  8. Gradshteyn, I. S. and I. M. Rhyzik (1965) Tables of Integrals, Series, and Products, Academic Press, New York.Google Scholar
  9. Johnson, N. L. and S. Kotz (1970) Continuous Univariate Distributions, vol. 1, John Wiley and Sons, New York.Google Scholar
  10. Kendall, M. G. and A. Stuart (1961) The Advanced Theory of Statistics, vol. 1, 2nd ed., Hafner, New York.Google Scholar
  11. Kloek, T. and H. K. van Dijk (1978) Efficient Estimation of Income Distribution Parameters, Journal or Econometrics, 8, 61-74.CrossRefGoogle Scholar
  12. McDonald, J. B. and B. C. Jensen (1979) An Analysis of Estimators of Alternative Measures of Income Inequality Associated with the Gamma Distribution Function, Journal of the American Statistical Association, 74, 856-860.CrossRefGoogle Scholar
  13. McDonald, J. B. and M. J. Ransom (1979) Functional Forms, Estimation Techniques and the Distribution of Income, Econometrica, 47, 1513-1526.CrossRefGoogle Scholar
  14. McDonald, J. B. and M. J. Ransom (1981) An Analysis of the Bounds for the Gini Coefficient, Journal of Econometrics, 17, 177-188.CrossRefGoogle Scholar
  15. Rainville, E. D. (1960) Special Functions, MacMillan, New York.Google Scholar
  16. Salem, A. B. Z. and T. D. Mount (1974) A Convenient Descriptive Model of Income Distribution: the Gamma Density, Econometrica, 42(6), 1115-1127.CrossRefGoogle Scholar
  17. Singh, S. K. and G. S. Maddala (1976) A Function for Size Distribution of Incomes, Econometrica, 44(5), 963-970.CrossRefGoogle Scholar
  18. Tadikamalla, P. R. (1980) A Look at the Burr and Related Distributions, International Statistical Review, 48, 337-349.CrossRefGoogle Scholar
  19. Taille, C. (1981) Lorenz Ordering Within the Generalized Gamma Family of Income Distributions, in G. P. Taille, C. and B. Balderssari (eds.) Statistical Distributions in Scientific Work, vol. 6, pp. 181-192, Reidel, Boston.Google Scholar
  20. Thurow, L. C. (1970) Analyzing the American Income Distribution, American Economic Review, 48, 261-269.Google Scholar
  21. (1961-1971) Income of Families and Persons in the United States, Current Population Reports, Series P-60, US Government Printing Office, Washington.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • James B. McDonald

There are no affiliations available

Personalised recommendations