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Estimating Lorenz Curves Using a Dirichlet Distribution

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  • Chotikapanich, D.
  • Griffiths, W.

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, estimation techniques that have good properties when 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 that recognises the cumulative proportional nature of the Lorenz curve data by assuming that the income proportions are distributed as a Dirichlet distribution. Five Lorenz-curve specifications are used to demonstrate the technique. Maximum likelihood estimates under the Dirichlet distribution assumption provide better-fitting Lorenz curves than nonlinear least squares and another estimation technique that has appeared in the literature.

Suggested Citation

  • Chotikapanich, D. & Griffiths, W., 2001. "Estimating Lorenz Curves Using a Dirichlet Distribution," Department of Economics - Working Papers Series 802, The University of Melbourne.
    • Handle: RePEc:mlb:wpaper:802
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    References listed on IDEAS

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    Cited by:

    1. Heshmati, Almas, 2004. "Inequalities and Their Measurement," IZA Discussion Papers 1219, Institute of Labor Economics (IZA).
    2. Xiaofeng Lv & Gupeng Zhang & Xinkuo Xu & Qinghai Li, 2017. "Bootstrap-calibrated empirical likelihood confidence intervals for the difference between two Gini indexes," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 15(2), pages 195-216, June.
    3. Gholamreza Hajargasht & William E. Griffiths, 2016. "Inference for Lorenz Curves," Department of Economics - Working Papers Series 2022, The University of Melbourne.
    4. Mathias Silva, 2023. "Parametric estimation of income distributions using grouped data: an Approximate Bayesian Computation approach [Working Papers / Documents de travail]," Working Papers hal-04066544, HAL.
    5. José M. R. Murteira & Joaquim J. S. Ramalho, 2016. "Regression Analysis of Multivariate Fractional Data," Econometric Reviews, Taylor & Francis Journals, vol. 35(4), pages 515-552, April.
    6. Genya Kobayashi & Kazuhiko Kakamu, 2019. "Approximate Bayesian computation for Lorenz curves from grouped data," Computational Statistics, Springer, vol. 34(1), pages 253-279, March.
    7. 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.
    8. Ashraf Gouda & Tamás Szántai, 2010. "On numerical calculation of probabilities according to Dirichlet distribution," Annals of Operations Research, Springer, vol. 177(1), pages 185-200, June.
    9. Wang, Dongliang & Zhao, Yichuan & Gilmore, Dirk W., 2016. "Jackknife empirical likelihood confidence interval for the Gini index," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 289-295.
    10. Chiara Gigliarano & Pietro Muliere, 2013. "Estimating the Lorenz curve and Gini index with right censored data: a Polya tree approach," METRON, Springer;Sapienza Università di Roma, vol. 71(2), pages 105-122, September.
    11. Hasegawa, Hikaru & Kozumi, Hideo, 2003. "Estimation of Lorenz curves: a Bayesian nonparametric approach," Journal of Econometrics, Elsevier, vol. 115(2), pages 277-291, August.
    12. Xiaofeng Lv & Gupeng Zhang & Xinkuo Xu & Qinghai Li, 2017. "Bootstrap-calibrated empirical likelihood confidence intervals for the difference between two Gini indexes," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 15(2), pages 195-216, June.
    13. Andrew C. Chang & Phillip Li & Shawn M. Martin, 2018. "Comparing cross‐country estimates of Lorenz curves using a Dirichlet distribution across estimators and datasets," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 33(3), pages 473-478, April.
    14. Heshmati, Almas, 2004. "A Review of Decomposition of Income Inequality," IZA Discussion Papers 1221, Institute of Labor Economics (IZA).
    15. Satya Paul & Sriram Shankar, 2020. "An alternative single parameter functional form for Lorenz curve," Empirical Economics, Springer, vol. 59(3), pages 1393-1402, September.
    16. Enora Belz, 2019. "Estimating Inequality Measures from Quantile Data," Economics Working Paper Archive (University of Rennes & University of Caen) 2019-09, Center for Research in Economics and Management (CREM), University of Rennes, University of Caen and CNRS.
    17. Khosravi Tanak, A. & Mohtashami Borzadaran, G.R. & Ahmadi, Jafar, 2018. "New functional forms of Lorenz curves by maximizing Tsallis entropy of income share function under the constraint on generalized Gini index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 511(C), pages 280-288.
    18. Enora Belz, 2019. "Estimating Inequality Measures from Quantile Data," Working Papers halshs-02320110, HAL.
    19. E. Gómez-Déniz, 2016. "A family of arctan Lorenz curves," Empirical Economics, Springer, vol. 51(3), pages 1215-1233, November.
    20. Xiaobo Shen & Pingsheng Dai, 2024. "A regression method for estimating Gini index by decile," Palgrave Communications, Palgrave Macmillan, vol. 11(1), pages 1-8, December.

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    More about this item

    Keywords

    Gini coefficient; maximum likelihood estimation;

    JEL classification:

    • C43 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Index Numbers and Aggregation
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

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