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Adjusted empirical likelihood method for the tail index of a heavy-tailed distribution

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  • Li, Yizeng
  • Qi, Yongcheng

Abstract

Empirical likelihood is a well-known nonparametric method in statistics and has been widely applied in statistical inference. The method has been employed by Lu and Peng (2002) to constructing confidence intervals for the tail index of a heavy-tailed distribution. It is demonstrated in Lu and Peng (2002) that the empirical likelihood-based confidence intervals perform better than confidence intervals based on normal approximation in terms of the coverage probability. In general, the empirical likelihood method can be hindered by its imprecision in the coverage probability when the sample size is small. This may cause a serious undercoverage issue when we apply the empirical likelihood to the tail index as only a very small portion of observations can be used in the estimation of the tail index. In this paper, we employ an adjusted empirical likelihood method, developed by Chen et al. (2008) and Liu and Chen (2010), to constructing confidence intervals of the tail index so as to achieve a better accuracy. We conduct a simulation study to compare the performance of the adjusted empirical likelihood method and the normal approximation method. Our simulation results indicate that the adjusted empirical likelihood method outperforms other methods in terms of the coverage probability and length of confidence intervals. We also apply the adjusted empirical likelihood method to a real data set.

Suggested Citation

  • Li, Yizeng & Qi, Yongcheng, 2019. "Adjusted empirical likelihood method for the tail index of a heavy-tailed distribution," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 50-58.
    • Handle: RePEc:eee:stapro:v:152:y:2019:i:c:p:50-58
    DOI: 10.1016/j.spl.2019.04.015
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    References listed on IDEAS

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    1. Tsao, Min, 2004. "A new method of calibration for the empirical loglikelihood ratio," Statistics & Probability Letters, Elsevier, vol. 68(3), pages 305-314, July.
    2. L. De Haan & L. Peng, 1998. "Comparison of tail index estimators," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 52(1), pages 60-70, March.
    3. Paul Embrechts & Sidney Resnick & Gennady Samorodnitsky, 1999. "Extreme Value Theory as a Risk Management Tool," North American Actuarial Journal, Taylor & Francis Journals, vol. 3(2), pages 30-41.
    4. McNeil, Alexander J., 1997. "Estimating the Tails of Loss Severity Distributions Using Extreme Value Theory," ASTIN Bulletin, Cambridge University Press, vol. 27(1), pages 117-137, May.
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