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Asymptotic behaviors of the Lorenz curve and Gini index in sampling from a length-biased distribution

Author

Listed:
  • Fakoor, V.
  • Ghalibaf, M. Bolbolian
  • Azarnoosh, H.A.

Abstract

In this work, we consider the nonparametric estimators of the Lorenz curve and Gini index based on a sample from the corresponding length-biased distribution. We show that this estimators are strongly consistent for the associated Lorenz curve and Gini index. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. We apply the strong Gaussian approximation technique to obtain a functional law for the iterated logarithm for the Lorenz curve. Also, we obtain an asymptotic normality for the corresponding Gini index.

Suggested Citation

  • Fakoor, V. & Ghalibaf, M. Bolbolian & Azarnoosh, H.A., 2011. "Asymptotic behaviors of the Lorenz curve and Gini index in sampling from a length-biased distribution," Statistics & Probability Letters, Elsevier, vol. 81(9), pages 1425-1435, September.
    • Handle: RePEc:eee:stapro:v:81:y:2011:i:9:p:1425-1435
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    References listed on IDEAS

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    1. Csörgö, Miklós & Zitikis, Ricardas, 1997. "On the rate of strong consistency of Lorenz curves," Statistics & Probability Letters, Elsevier, vol. 34(2), pages 113-121, June.
    2. Gastwirth, Joseph L, 1971. "A General Definition of the Lorenz Curve," Econometrica, Econometric Society, vol. 39(6), pages 1037-1039, November.
    3. Csörgo, Miklós & Zitikis, Ricardas, 1996. "Strassen's LIL for the Lorenz Curve," Journal of Multivariate Analysis, Elsevier, vol. 59(1), pages 1-12, October.
    4. Davydov, Youri & Zitikis, Ricardas, 2002. "Convergence of generalized Lorenz curves based on stationary ergodic random sequences with deterministic noise," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 329-340, October.
    5. Sze-Man Tse, 2006. "Lorenz Curve for Truncated and Censored Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(4), pages 675-686, December.
    6. Rao, C. R., 1995. "Strassen's Law of the Iterated Logarithm for the Lorenz Curves," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 239-252, August.
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