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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.

Suggested Citation

  • Schluter, Christian & van Garderen, Kees Jan, 2009. "Edgeworth expansions and normalizing transforms for inequality measures," Journal of Econometrics, Elsevier, vol. 150(1), pages 16-29, May.
  • Handle: RePEc:eee:econom:v:150:y:2009:i:1:p:16-29
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    1. Margherita Gerolimetto & Stefano Magrini, 2018. "Inference for inequality measures: a review," RIEDS - Rivista Italiana di Economia, Demografia e Statistica - The Italian Journal of Economic, Demographic and Statistical Studies, SIEDS Societa' Italiana di Economia Demografia e Statistica, vol. 72(2), pages 75-85, April-Jun.
    2. Jean-Marie Dufour & Emmanuel Flachaire & Lynda Khalaf, 2019. "Permutation Tests for Comparing Inequality Measures," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(3), pages 457-470, July.
    3. Abul Naga, Ramses H. & Shen, Yajie & Yoo, Hong Il, 2016. "Joint hypothesis tests for multidimensional inequality indices," Economics Letters, Elsevier, vol. 141(C), pages 138-142.
    4. Frank Cowell & Emmanuel Flachaire & Sanghamitra Bandyopadhyay, 2013. "Reference distributions and inequality measurement," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 11(4), pages 421-437, December.
    5. Silvia De Nicol`o & Maria Rosaria Ferrante & Silvia Pacei, 2021. "Mind the Income Gap: Behavior of Inequality Estimators from Complex Survey Small Samples," Papers 2107.08950, arXiv.org, revised Jul 2021.
    6. Stéphane Guerrier & Samuel Orso & Maria-Pia Victoria-Feser, 2018. "Parametric Inference for Index Functionals," Econometrics, MDPI, vol. 6(2), pages 1-11, April.
    7. Frank A. Cowell & Philippe Kerm, 2015. "Wealth Inequality: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 29(4), pages 671-710, September.

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