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A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator

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  • Gomes, M. Ivette
  • Pestana, Dinis
  • Caeiro, Frederico

Abstract

For heavy tails, with a positive tail index [gamma], classical tail index estimators, like the Hill estimator, are known to be quite sensitive to the number of top order statistics k used in the estimation, whereas second-order reduced-bias estimators show much less sensitivity to changes in k. In the recent minimum-variance reduced-bias (MVRB) tail index estimators, the estimation of the second order parameters in the bias has been performed at a level k1 of a larger order than that of the level k at which we compute the tail index estimators. Such a procedure enables us to keep the asymptotic variance of the new estimators equal to the asymptotic variance of the Hill estimator, for all k at which we can guarantee the asymptotic normality of the Hill statistics. These values of k, as well as larger values of k, will also enable us to guarantee the asymptotic normality of the reduced-bias estimators, but, to reach the minimal mean squared error of these MVRB estimators, we need to work with levels k and k1 of the same order. In this note we derive the way the asymptotic variance varies as a function of q, the finite limiting value of k/k1, as the sample size n increases to infinity.

Suggested Citation

  • Gomes, M. Ivette & Pestana, Dinis & Caeiro, Frederico, 2009. "A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator," Statistics & Probability Letters, Elsevier, vol. 79(3), pages 295-303, February.
  • Handle: RePEc:eee:stapro:v:79:y:2009:i:3:p:295-303
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    References listed on IDEAS

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    1. Gomes, M. Ivette & Pestana, Dinis, 2007. "A Sturdy Reduced-Bias Extreme Quantile (VaR) Estimator," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 280-292, March.
    2. Liang Peng & Yongcheng Qi, 2004. "Estimating the First‐ and Second‐Order Parameters of a Heavy‐Tailed Distribution," Australian & New Zealand Journal of Statistics, Australian Statistical Publishing Association Inc., vol. 46(2), pages 305-312, June.
    3. M. Ivette Gomes & Laurens De Haan & Lígia Henriques Rodrigues, 2008. "Tail index estimation for heavy‐tailed models: accommodation of bias in weighted log‐excesses," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 31-52, February.
    4. 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.
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    2. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2021. "ExpectHill estimation, extreme risk and heavy tails," Journal of Econometrics, Elsevier, vol. 221(1), pages 97-117.
    3. El Methni, Jonathan & Stupfler, Gilles, 2018. "Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions," Econometrics and Statistics, Elsevier, vol. 6(C), pages 129-148.
    4. Tertius de Wet & Yuri Goegebeur & Armelle Guillou, 2012. "Weighted Moment Estimators for the Second Order Scale Parameter," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 753-783, September.

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