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Edgeworth expansions and normalizing transforms for inequality measures

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  • Schluter, Christian
  • van Garderen, Kees Jan

Abstract

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second-order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second-order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n-1). This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n-3/2). We illustrate our procedure with an application to regional inequality measurement in Côte d'Ivoire.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Econometrics.

Volume (Year): 150 (2009)
Issue (Month): 1 (May)
Pages: 16-29

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Handle: RePEc:eee:econom:v:150:y:2009:i:1:p:16-29

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Web page: http://www.elsevier.com/locate/jeconom

Related research

Keywords: Generalized Entropy inequality measures Higher- order expansions Normalizing transformations;

References

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  1. Kloek, Teun & van Dijk, Herman K., 1978. "Efficient estimation of income distribution parameters," Journal of Econometrics, Elsevier, vol. 8(1), pages 61-74, August.
  2. Russell Davidson & Emmanuel Flachaire, 2004. "Asymptotic and bootstrap inference for inequality and poverty measures," Cahiers de la Maison des Sciences Economiques v04100, Université Panthéon-Sorbonne (Paris 1).
  3. Schluter, Christian & Trede, Mark, 2002. "Tails of Lorenz curves," Journal of Econometrics, Elsevier, vol. 109(1), pages 151-166, July.
  4. McDonald, James B, 1984. "Some Generalized Functions for the Size Distribution of Income," Econometrica, Econometric Society, vol. 52(3), pages 647-63, May.
  5. Cowell, F.A., 2000. "Measurement of inequality," Handbook of Income Distribution, in: A.B. Atkinson & F. Bourguignon (ed.), Handbook of Income Distribution, edition 1, volume 1, chapter 2, pages 87-166 Elsevier.
  6. Marsh, Patrick, 2004. "Transformations For Multivariate Statistics," Econometric Theory, Cambridge University Press, vol. 20(05), pages 963-987, October.
  7. Cowell, Frank A, 1980. "On the Structure of Additive Inequality Measures," Review of Economic Studies, Wiley Blackwell, vol. 47(3), pages 521-31, April.
  8. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
  9. Mills, Jeffrey A & Zandvakili, Sourushe, 1997. "Statistical Inference via Bootstrapping for Measures of Inequality," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 12(2), pages 133-50, March-Apr.
  10. Thistle, Paul D, 1990. "Large Sample Properties of Two Inequality Indices," Econometrica, Econometric Society, vol. 58(3), pages 725-28, May.
  11. DAVIDSON, Russell & DUCLOS, Jean-Yves, 1995. "Statistical Inference for the Measurement of the Incidences of Taxes and Transfers," Cahiers de recherche 9521, Université Laval - Département d'économique.
  12. Cowell, Frank A., 1989. "Sampling variance and decomposable inequality measures," Journal of Econometrics, Elsevier, vol. 42(1), pages 27-41, September.
  13. Singh, S K & Maddala, G S, 1976. "A Function for Size Distribution of Incomes," Econometrica, Econometric Society, vol. 44(5), pages 963-70, September.
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Cited by:
  1. Frank Cowell & Emmanuel Flachaire & Sanghamitra Bandyopadhyay, 2013. "Reference distributions and inequality measurement," Journal of Economic Inequality, Springer, vol. 11(4), pages 421-437, December.

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