IDEAS home Printed from https://ideas.repec.org/a/taf/emetrv/v26y2007i5p567-577.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/07474930701512170
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1080/07474930701512170?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. W. Sendler, 1979. "On statistical inference in concentration measurement," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 26(1), pages 109-122, December.
    3. 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.
    4. 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.
    5. James E. Foster & Efe A. Ok, 1999. "Lorenz Dominance and the Variance of Logarithms," Econometrica, Econometric Society, vol. 67(4), pages 901-908, July.
    6. Tomson Ogwang, 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.
    7. Kuan Xu & Lars Osberg, 2002. "The social welfare implications, decomposability, and geometry of the Sen family of poverty indices," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 35(1), pages 138-152, February.
    8. Elias Karagiannis & Milorad Kovacevic', 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.
    9. 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.
    10. 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.
    11. repec:bla:obuest:v:62:y:2000:i:1:p:123-29 is not listed on IDEAS
    12. repec:bla:obuest:v:62:y:2000:i:1:p:119-22 is not listed on IDEAS
    13. Sen, Amartya, 1973. "On Economic Inequality," OUP Catalogue, Oxford University Press, number 9780198281931.
    14. 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.
    15. David E. A. Giles, 2004. "Calculating a Standard Error for the Gini Coefficient: Some Further Results," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 66(3), pages 425-433, July.
    16. Satya R. Chakravarty, 2001. "Why Measuring Inequality by the Variance Makes Sense from a Theoretical Point of View," Journal of Income Distribution, Ad libros publications inc., vol. 10(3-4), pages 6-6, September.
    17. Lars Osberg, 2000. "Poverty in Canada and the United States: measurement, trends, and implications," Canadian Journal of Economics, Canadian Economics Association, vol. 33(4), pages 847-877, November.
    18. Sen, Amartya K, 1976. "Poverty: An Ordinal Approach to Measurement," Econometrica, Econometric Society, vol. 44(2), pages 219-231, March.
    19. 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.
    20. 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.
    21. Zheng, Buhong, et al, 2000. "Inequality Orderings, Normalized Stochastic Dominance, and Statistical Inference," Journal of Business & Economic Statistics, American Statistical Association, vol. 18(4), pages 479-488, October.
    22. Biewen, Martin, 2002. "Bootstrap inference for inequality, mobility and poverty measurement," Journal of Econometrics, Elsevier, vol. 108(2), pages 317-342, June.
    23. 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 113-119, January.
    24. 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, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Russell Davidson, 2010. "Innis Lecture: Inference on income distributions," Canadian Journal of Economics/Revue canadienne d'économique, John Wiley & Sons, vol. 43(4), pages 1122-1148, November.
    3. Ilaria Benedetti & Federico Crescenzi & Tiziana Laureti, 2020. "Measuring Uncertainty for Poverty Indicators at Regional Level: The Case of Mediterranean Countries," Sustainability, MDPI, vol. 12(19), pages 1-19, October.
    4. Francesco Andreoli & Eugenio Peluso, 2016. "So close yet so unequal: Reconsidering spatial inequality in U.S. cities," Working Papers 21/2016, University of Verona, Department of Economics.
    5. Laurens Cherchye & Thomas Demuynck & Bram De Rock, 2013. "Nash‐Bargained Consumption Decisions: A Revealed Preference Analysis," Economic Journal, Royal Economic Society, vol. 123, pages 195-235, March.
    6. Francesco Andreoli & Eugenio Peluso, 2021. "Inference for the neighbourhood inequality index," Spatial Economic Analysis, Taylor & Francis Journals, vol. 16(3), pages 313-332, July.
    7. 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.
    8. 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.
    9. Davidson, Russell, 2009. "Reliable inference for the Gini index," Journal of Econometrics, Elsevier, vol. 150(1), pages 30-40, May.
    10. Philippe De Vreyer & Sylvie Lambert, 2021. "Inequality, Poverty, and the Intra-Household Allocation of Consumption in Senegal," The World Bank Economic Review, World Bank, vol. 35(2), pages 414-435.
    11. Yang Wei & Zhouping Li & Yunqiu Dai, 2022. "Unified smoothed jackknife empirical likelihood tests for comparing income inequality indices," Statistical Papers, Springer, vol. 63(5), pages 1415-1475, October.
    12. Bhargab Chattopadhyay & Shyamal Krishna De, 2016. "Estimation of Gini Index within Pre-Specified Error Bound," Econometrics, MDPI, vol. 4(3), pages 1-12, June.
    13. ANDREOLI Francesco & PELUSO Eugenio, 2017. "So close yet so unequal: Spatial inequality in American cities," LISER Working Paper Series 2017-11, Luxembourg Institute of Socio-Economic Research (LISER).
    14. Yoonseok Lee & Donggyun Shin, 2016. "Measuring Social Tension from Income Class Segregation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(3), pages 457-471, July.
    15. 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.
    16. 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.
    17. Shyamal K. De & Bhargab Chattopadhyay, 2017. "Minimum Risk Point Estimation of Gini Index," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 247-277, November.
    18. Yong Tao & Xiangjun Wu & Changshuai Li, 2014. "Rawls' Fairness, Income Distribution and Alarming Level of Gini Coefficient," Papers 1409.3979, arXiv.org.
    19. Sudheesh K. Kattumannil & N. Sreelakshmi & N. Balakrishnan, 2022. "Non-Parametric Inference for Gini Covariance and its Variants," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 790-807, August.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Davidson, Russell, 2009. "Reliable inference for the Gini index," Journal of Econometrics, Elsevier, vol. 150(1), pages 30-40, May.
    2. 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.
    3. Xiaofeng Lv & Gupeng Zhang & Guangyu Ren, 2017. "Gini index estimation for lifetime data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(2), pages 275-304, April.
    4. Karoly, Lynn & Schröder, Carsten, 2015. "Fast methods for jackknifing inequality indices," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 37(1), pages 125-138.
    5. David E. A. Giles, 2004. "Calculating a Standard Error for the Gini Coefficient: Some Further Results," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 66(3), pages 425-433, July.
    6. 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.
    7. 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.
    8. Kuan Xu & Ian Irvine, 2002. "Crime, Punishment and the Measurement of Poverty in the United States, 1979-1997," LIS Working papers 333, LIS Cross-National Data Center in Luxembourg.
    9. Berger Yves G. & Balay İklim Gedik, 2020. "Confidence Intervals of Gini Coefficient Under Unequal Probability Sampling," Journal of Official Statistics, Sciendo, vol. 36(2), pages 237-249, June.
    10. Stephen P. Jenkins & John Micklewright, 2007. "New Directions in the Analysis of Inequality and Poverty," Discussion Papers of DIW Berlin 700, DIW Berlin, German Institute for Economic Research.
    11. Picot, Garnett & Morissette, Rene & Myles, John, 2003. "Low-income Intensity During the 1990s: The Role of Economic Growth, Employment Earnings and Social Transfers," Analytical Studies Branch Research Paper Series 2003172e, Statistics Canada, Analytical Studies Branch.
    12. 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.
    13. Qin, Yongsong & Rao, J.N.K. & Wu, Changbao, 2010. "Empirical likelihood confidence intervals for the Gini measure of income inequality," Economic Modelling, Elsevier, vol. 27(6), pages 1429-1435, November.
    14. 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.
    15. Makdissi, Paul & Groleau, Yves, 2002. "Que pouvons-nous apprendre des profils de pauvreté canadiens?," L'Actualité Economique, Société Canadienne de Science Economique, vol. 78(2), pages 257-286, Juin.
    16. WANG, Zuxiang & SMYTH, Russell & NG, Yew-Kwang, 2009. "A new ordered family of Lorenz curves with an application to measuring income inequality and poverty in rural China," China Economic Review, Elsevier, vol. 20(2), pages 218-235, June.
    17. Wen-Hao Chen & Jean-Yves Duclos, 2011. "Testing for poverty dominance: an application to Canada," Canadian Journal of Economics, Canadian Economics Association, vol. 44(3), pages 781-803, August.
    18. Mazzi Gian Luigi & Mitchell James & Carausu Florabela, 2021. "Measuring and Communicating the Uncertainty in Official Economic Statistics," Journal of Official Statistics, Sciendo, vol. 37(2), pages 289-316, June.
    19. Zheng, Buhong & J. Cushing, Brian, 2001. "Statistical inference for testing inequality indices with dependent samples," Journal of Econometrics, Elsevier, vol. 101(2), pages 315-335, April.
    20. Lingsheng Meng & Binzhen Wu & Zhaoguo Zhan, 2016. "Linear regression with an estimated regressor: applications to aggregate indicators of economic development," Empirical Economics, Springer, vol. 50(2), pages 299-316, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:emetrv:v:26:y:2007:i:5:p:567-577. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: http://www.tandfonline.com/LECR20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.