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U-Statistics and Their Asymptotic Results for Some Inequality and Poverty Measures

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  • Kuan Xu

Abstract

U-statistics form a general class of statistics that have certain important features in common. This class arises as a generalization of the sample mean and the sample variance, and typically members of the class are asymptotically normal with good consistency properties. The class encompasses some widely used income inequality and poverty measures, in particular the variance, the Gini index, the poverty rate, the average poverty gap ratios, the Foster-Greer-Thorbecke index, and the Sen index and its modified form. This paper illustrates how these measures come together within the class of U-statistics, and thereby why U-statistics are useful in econometrics.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:emetrv:v:26:y:2007:i:5:p:567-577
    DOI: 10.1080/07474930701512170
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    1. Lars Osberg & Kuan Xu, 1999. "Poverty Intensity: How Well Do Canadian Provinces Compare?," Canadian Public Policy, University of Toronto Press, vol. 25(2), pages 179-195, June.
    2. Kuan Xu & Lars Osberg, 2002. "The social welfare implications, decomposability, and geometry of the Sen family of poverty indices," Canadian Journal of Economics, Canadian Economics Association, vol. 35(1), pages 138-152, February.
    3. Bishop, John A & Chakraborti, S & Thistle, Paul D, 1990. "An Asymptotically Distribution-Free Test for Sen's Welfare Index," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 52(1), pages 105-113, February.
    4. 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.
    5. Yitzhaki, Shlomo, 1991. "Calculating Jackknife Variance Estimators for Parameters of the Gini Method," Journal of Business & Economic Statistics, American Statistical Association, vol. 9(2), pages 235-239, April.
    6. repec:jid:journl:y:2001:v:10:i:3-4:p:6-6 is not listed on IDEAS
    7. Sen, Amartya K, 1976. "Poverty: An Ordinal Approach to Measurement," Econometrica, Econometric Society, vol. 44(2), pages 219-231, March.
    8. Kuan Xu, 1999. "Statistical Inference for the Sen-Shorrocks-Thon Index of Poverty Intensity," Journal of Income Distribution, Ad libros publications inc., vol. 8(1), pages 8-8, June.
    9. Bishop, John A & Formby, John P & Zheng, Buhong, 1998. "Inference Tests for Gini-Based Tax Progressivity Indexes," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 322-330, July.
    10. James E. Foster & Efe A. Ok, 1999. "Lorenz Dominance and the Variance of Logarithms," Econometrica, Econometric Society, vol. 67(4), pages 901-908, July.
    11. 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 123-129, February.
    12. Karagiannis, Elias & Kovacevic', Milorad, 2000. " A Method to Calculate the Jackknife Variance Estimator for the Gini Coefficient," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 62(1), pages 119-122, February.
    13. Sen, Amartya, 1973. "On Economic Inequality," OUP Catalogue, Oxford University Press, number 9780198281931.
    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 381-387, May.
    15. Lars Osberg & Kuan Xu, 2000. "International Comparisons of Poverty Intensity: Index Decomposition and Bootstrap Inference," Journal of Human Resources, University of Wisconsin Press, vol. 35(1), pages 51-81.
    16. Biewen, Martin, 2002. "Bootstrap inference for inequality, mobility and poverty measurement," Journal of Econometrics, Elsevier, vol. 108(2), pages 317-342, June.
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    Citations

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    Cited by:

    1. Thomas Demuynck, 2012. "An (almost) unbiased estimator for the S-Gini index," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 10(1), pages 109-126, March.
    2. Yoonseok Lee & Donggyun Shin, 2013. "Measuring Social Unrest Based on Income Distribution," Center for Policy Research Working Papers 160, Center for Policy Research, Maxwell School, Syracuse University.
    3. Davidson, Russell, 2009. "Reliable inference for the Gini index," Journal of Econometrics, Elsevier, vol. 150(1), pages 30-40, May.
    4. Bhargab Chattopadhyay & Shyamal Krishna De, 2016. "Estimation of Gini Index within Pre-Specified Error Bound," Econometrics, MDPI, Open Access Journal, vol. 4(3), pages 1-12, June.
    5. ANDREOLI Francesco & PELUSO Eugenio, 2017. "So close yet so unequal: Spatial inequality in American cities," LISER Working Paper Series 2017-11, LISER.
    6. Yong Tao & Xiangjun Wu & Changshuai Li, 2014. "Rawls' Fairness, Income Distribution and Alarming Level of Gini Coefficient," Papers 1409.3979, arXiv.org.
    7. 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.
    8. repec:taf:jnlbes:v:34:y:2016:i:3:p:457-471 is not listed on IDEAS
    9. ANDREOLI Francesco, 2018. "Inference for the neighborhood inequality index," LISER Working Paper Series 2018-19, LISER.
    10. Philippe de Vreyer & Sylvie Lambert, 2019. "Inequality, poverty and the intra-household allocation of consumption in Senegal," PSE Working Papers halshs-02177745, HAL.
    11. James Foster & Joel Greer & Erik Thorbecke, 2010. "The Foster–Greer–Thorbecke (FGT) poverty measures: 25 years later," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 8(4), pages 491-524, December.
    12. James E. Foster & Joel Greer & Erik Thorbecke, 2010. "The Foster-Greer-Thorbecke (FGT) Poverty Measures: Twenty-Five Years Later," Working Papers 2010-14, The George Washington University, Institute for International Economic Policy.
    13. Sylvie Lambert & Philippe De Vreyer, 2017. "By ignoring intra-household inequality do we underestimate the extent of poverty?," Working Papers DT/2017/05, DIAL (Développement, Institutions et Mondialisation).
    14. repec:spr:sankhb:v:79:y:2017:i:2:d:10.1007_s13571-017-0140-3 is not listed on IDEAS

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