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Estimating Lorenz Curves Using a Dirichlet Distribution

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Author Info
Duangkamon Chotikapanich (Curtin University of Technology)
William E. Griffiths (University of New England)

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Abstract

The Lorenz curve relates the cumulative proportion of income to the cumulative proportion of population. When a particular functional form of the Lorenz curve is specified it is typically estimated by linear or nonlinear least squares assuming that the error terms are independently and normally distributed. Observations on cumulative proportions are clearly neither independent nor normally distributed. This paper proposes and applies a new methodology which recognizes the cumulative proportional nature of the Lorenz curve data by assuming that the proportion of income is distributed as a Dirichlet distribution. Five Lorenz-curve specifications were used to demonstrate the technique. Once a likelihood function and the posterior probability density function for each specification are derived we can use maximum likelihood or Bayesian estimation to estimate the parameters. Maximum likelihood estimates and Bayesian posterior probability density functions for the Gini coefficient are also obtained for each Lorenz-curve specification.

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Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 1215.

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Date of creation: 01 Aug 2000
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Handle: RePEc:ecm:wc2000:1215

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Newey, Whitney K & West, Kenneth D, 1987. "A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica, Econometric Society, vol. 55(3), pages 703-08, May. [Downloadable!] (restricted)
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  2. Ryu, Hang K. & Slottje, Daniel J., 1996. "Two flexible functional form approaches for approximating the Lorenz curve," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 251-274. [Downloadable!] (restricted)
  3. Kakwani, N C & Podder, N, 1973. "On the Estimation of Lorenz Curves from Grouped Observations," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 14(2), pages 278-92, June. [Downloadable!] (restricted)
  4. Kakwani, Nanak C & Podder, N, 1976. "Efficient Estimation of the Lorenz Curve and Associated Inequality Measures from Grouped Observations," Econometrica, Econometric Society, vol. 44(1), pages 137-48, January. [Downloadable!] (restricted)
  5. Chotikapanich, Duangkamon, 1993. "A comparison of alternative functional forms for the Lorenz curve," Economics Letters, Elsevier, vol. 41(2), pages 129-138. [Downloadable!] (restricted)
  6. Sarabia, J. -M. & Castillo, Enrique & Slottje, Daniel J., 1999. "An ordered family of Lorenz curves," Journal of Econometrics, Elsevier, vol. 91(1), pages 43-60, July. [Downloadable!] (restricted)
  7. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 69(2), pages 427-428, October. [Downloadable!] (restricted)
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  8. Gastwirth, Joseph L, 1972. "The Estimation of the Lorenz Curve and Gini Index," The Review of Economics and Statistics, MIT Press, vol. 54(3), pages 306-16, August. [Downloadable!] (restricted)
  9. Beach, Charles M & Davidson, Russell, 1983. "Distribution-Free Statistical Inference with Lorenz Curves and Income Shares," Review of Economic Studies, Blackwell Publishing, vol. 50(4), pages 723-35, October. [Downloadable!] (restricted)
  10. Woodland, A. D., 1979. "Stochastic specification and the estimation of share equations," Journal of Econometrics, Elsevier, vol. 10(3), pages 361-383, August. [Downloadable!] (restricted)
  11. Rasche, R H, et al, 1980. "Functional Forms for Estimating the Lorenz Curve: Comment," Econometrica, Econometric Society, vol. 48(4), pages 1061-62, May. [Downloadable!] (restricted)
  12. Bishop, John A & Chakraborti, S & Thistle, Paul D, 1989. "Asymptotically Distribution-Free Statistical Inference for Generalized Lorenz Curves," The Review of Economics and Statistics, MIT Press, vol. 71(4), pages 725-27, November. [Downloadable!] (restricted)
  13. McDonald, James B, 1984. "Some Generalized Functions for the Size Distribution of Income," Econometrica, Econometric Society, vol. 52(3), pages 647-63, May. [Downloadable!] (restricted)
  14. Datt, Gaurav, 1998. "Computational tools for poverty measurement and analysis," FCND discussion papers 50, International Food Policy Research Institute (IFPRI). [Downloadable!]
  15. Shorrocks, Anthony F, 1983. "Ranking Income Distributions," Economica, London School of Economics and Political Science, vol. 50(197), pages 3-17, February. [Downloadable!] (restricted)
  16. Kakwani, Nanak, 1980. "On a Class of Poverty Measures," Econometrica, Econometric Society, vol. 48(2), pages 437-46, March. [Downloadable!] (restricted)
  17. Basmann, R. L. & Hayes, K. J. & Slottje, D. J. & Johnson, J. D., 1990. "A general functional form for approximating the Lorenz curve," Journal of Econometrics, Elsevier, vol. 43(1-2), pages 77-90. [Downloadable!] (restricted)
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Heshmati, Almas, 2004. "Inequalities and Their Measurement," IZA Discussion Papers 1219, Institute for the Study of Labor (IZA). [Downloadable!]
  2. Heshmati, Almas, 2004. "A Review of Decomposition of Income Inequality," IZA Discussion Papers 1221, Institute for the Study of Labor (IZA). [Downloadable!]
  3. T. Kämpke & R. Pestel & F.J. Radermacher, 2003. "A Computational Concept for Normative Equity," European Journal of Law and Economics, Springer, vol. 15(2), pages 129-163, March. [Downloadable!] (restricted)
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