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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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    1. 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.
    2. 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.
    3. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    4. 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 278-292, June.
    5. Rasche, R H, et al, 1980. "Functional Forms for Estimating the Lorenz Curve: Comment," Econometrica, Econometric Society, vol. 48(4), pages 1061-1062, May.
    6. 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 137-148, January.
    7. 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 306-316, August.
    8. 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.
    9. 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 725-727, November.
    10. Kakwani, Nanak, 1980. "On a Class of Poverty Measures," Econometrica, Econometric Society, vol. 48(2), pages 437-446, March.
    11. repec:bla:econom:v:50:y:1983:i:197:p:3-17 is not listed on IDEAS
    12. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 33(1), pages 125-132.
    13. Datt, Gaurav, 1998. "Computational tools for poverty measurement and analysis," FCND discussion papers 50, International Food Policy Research Institute (IFPRI).
    14. 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.
    15. Chotikapanich, Duangkamon, 1993. "A comparison of alternative functional forms for the Lorenz curve," Economics Letters, Elsevier, vol. 41(2), pages 129-138.
    16. Woodland, A. D., 1979. "Stochastic specification and the estimation of share equations," Journal of Econometrics, Elsevier, vol. 10(3), pages 361-383, August.
    17. Charles M. Beach & Russell Davidson, 1983. "Distribution-Free Statistical Inference with Lorenz Curves and Income Shares," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 50(4), pages 723-735.
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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.

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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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