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Reliable inference for the GINI Index

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  • Russell Davidson

    ()
    (GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille - Université de la Méditerranée - Aix-Marseille II - Université Paul Cézanne - Aix-Marseille III - Ecole des Hautes Etudes en Sciences Sociales (EHESS) - CNRS : UMR6579, CIREG - Centre interuniversitaire de recherche en économie quantitative - Université de Montréal, Department of Economics, McGill University - McGill University)

Abstract

Although attention has been given to obtaining reliable standard errors for the plugin estimator of the Gini index, all standard errors suggested until now are either complicated or quite unreliable. An approximation is derived for the estimator by which it is expressed as a sum of IID random variables. This approximation allows us to develop a reliable standard error that is simple to compute. A simple but effective bias correction is also derived. The quality of inference based on the approximation is checked in a number of simulation experiments, and is found to be very good unless the tail of the underlying distribution is heavy. Bootstrap methods are presented which alleviate this problem except in cases in which the variance is very large or fails to exist. Similar methods can be used to find reliable standard errors of other indices which are not simply linear functionals of the distribution function, such as Sen's poverty index and its modification known as the Sen-Shorrocks-Thon index.

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Bibliographic Info

Paper provided by HAL in its series Working Papers with number halshs-00443553.

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Date of creation: 30 Dec 2009
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Handle: RePEc:hal:wpaper:halshs-00443553

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Keywords: Gini index; delta method; asymptotic inference; jackknife; bootstrap;

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References

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  1. Tomson Ogwang, 2006. "A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality: Comment," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 68(3), pages 391-393, 06.
  2. Russell Davidson & James G. MacKinnon, 2001. "Bootstrap Tests: How Many Bootstraps?," Working Papers 1036, Queen's University, Department of Economics.
  3. Donaldson, David & Weymark, John A., 1980. "A single-parameter generalization of the Gini indices of inequality," Journal of Economic Theory, Elsevier, vol. 22(1), pages 67-86, February.
  4. David E. A. Giles, 2002. "Calculating a Standard Error for the Gini Coefficient: Some Further Results," Econometrics Working Papers 0202, Department of Economics, University of Victoria.
  5. Kuan Xu, 2007. "U-Statistics and Their Asymptotic Results for Some Inequality and Poverty Measures," Econometric Reviews, Taylor & Francis Journals, vol. 26(5), pages 567-577.
  6. George Deltas, 2003. "The Small-Sample Bias of the Gini Coefficient: Results and Implications for Empirical Research," The Review of Economics and Statistics, MIT Press, vol. 85(1), pages 226-234, February.
  7. Sen, Amartya K, 1976. "Poverty: An Ordinal Approach to Measurement," Econometrica, Econometric Society, vol. 44(2), pages 219-31, March.
  8. Jeffrey A. Mills & Sourushe Zandvakili, 1999. "Statistical Inference via Bootstrapping for Measures of Inequality," Macroeconomics 9902003, EconWPA.
  9. Tomson Ogwang, 2004. "Calculating a Standard Error for the Gini Coefficient: Some Further Results: Reply," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 66(3), pages 435-437, 07.
  10. Sen, Amartya, 1973. "On Economic Inequality," OUP Catalogue, Oxford University Press, number 9780198281931, October.
  11. Reza Modarres & Joseph L. Gastwirth, 2006. "A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 68(3), pages 385-390, 06.
  12. Shorrocks, Anthony F, 1995. "Revisiting the Sen Poverty Index," Econometrica, Econometric Society, vol. 63(5), pages 1225-30, September.
  13. Sandstrom, Arne & Wretman, Jan H & Walden, Bertil, 1988. "Variance Estimators of the Gini Coefficient--Probability Sampling," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(1), pages 113-19, January.
  14. Bishop, John A & Formby, John P & Zheng, Buhong, 1997. "Statistical Inference and the Sen Index of Poverty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(2), pages 381-87, May.
  15. Ogwang, Tomson, 2000. " A Convenient Method of Computing the Gini Index and Its Standard Error," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 62(1), pages 123-29, February.
  16. David E. A. Giles, 2006. "A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality: Comment," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 68(3), pages 395-396, 06.
  17. Bhattacharya, Debopam, 2007. "Inference on inequality from household survey data," Journal of Econometrics, Elsevier, vol. 137(2), pages 674-707, April.
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Citations

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Cited by:
  1. William E. Griffiths and Gholamreza Hajargasht, 2012. "GMM Estimation of Mixtures from Grouped Data:," Department of Economics - Working Papers Series 1148, The University of Melbourne.
  2. Stéphane Mussard & Patrick Richard, 2008. "Linking Yitzhaki’s and Dagum’s Gini Decompositions," Cahiers de recherche 08-21, Departement d'Economique de la Faculte d'administration à l'Universite de Sherbrooke.
  3. T. Demuynck, 2009. "An (almost) unbiased estimator for the S-Gini index," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 09/569, Ghent University, Faculty of Economics and Business Administration.
  4. Luis Fernando Gamboa & Andrés García-Suaza & Jesús Otero, 2010. "Statistical inference for testing Gini Coefficients: An application for Colombia," ENSAYOS SOBRE POLÍTICA ECONÓMICA, BANCO DE LA REPÚBLICA - ESPE.
  5. Judith A. Clarke & Ahmed A. Hoque, 2014. "On Variance Estimation for a Gini Coefficient Estimator Obtained from Complex Survey Data," Econometrics Working Papers 1401, Department of Economics, University of Victoria.
  6. Russell Davidson, 2010. "An Agnostic Look at Bayesian Statistics and Econometrics," Review of Economic Analysis, Rimini Centre for Economic Analysis, vol. 2(2), pages 153-168, June.
  7. Judith Clarke & Nilanjana Roy, 2012. "On statistical inference for inequality measures calculated from complex survey data," Empirical Economics, Springer, vol. 43(2), pages 499-524, October.
  8. Alina Jędrzejczak, 2012. "Estimation of Concentration Measures and Their Standard Errors for Income Distributions in Poland," International Advances in Economic Research, Springer, vol. 18(3), pages 287-297, August.

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