Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 924013, 16 pages
http://dx.doi.org/10.1155/2010/924013
Research Article

Inference about the Tail of a Distribution: Improvementon the Hill Estimator

Department of Theoretical and Mathematical Physics, Université de Mons, 20 Place du Parc, 7000 Mons, Belgium

Received 7 July 2009; Revised 14 March 2010; Accepted 15 April 2010

Academic Editor: Kenneth Berenhaut

Copyright © 2010 Jean Nuyts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality,” Physical Review A, vol. 38, no. 1, pp. 364–374, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  2. V. Pareto, Cours d'Économie Politique, Droz, Geneva, Switzerland, 1896.
  3. V. Pareto, Cours d'Économie Politique, Rouge, Paris, France, 1897.
  4. G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949.
  5. B. B. Mandelbrot, Fractals and Scaling in Finance, Springer, New York, NY, USA, 1997. View at MathSciNet
  6. M. Shlesinger, G. M. Zaslavsky, and U. Frisch, Lévy Flights and Related Topics in Physics, Springer, New York, NY, USA, 1995.
  7. J. Nuyts and I. Platten, “Phenomenology of the term structure of interest rates with padé approximants,” Physica A, vol. 299, pp. 528–546, 2001. View at Publisher · View at Google Scholar
  8. T. Alderweireld and J. Nuyts, “Detailed empirical study of the term structure of interest rates. Emergence of power laws and scaling laws,” Physica A, vol. 331, pp. 602–616, 2003, cond-mat/0305689. View at Google Scholar
  9. T. Alderweireld and J. Nuyts, “A theory for the term structure of interest rates,” cond-mat/0405293.
  10. A. Rényi, “On the theory of order statistics,” Acta Mathematica Academiae Scientiarum Hungaricae, vol. 4, pp. 191–231, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. B. M. Hill, “A simple general approach to inference about the tail of a distribution,” The Annals of Statistics, vol. 3, no. 5, pp. 1163–1174, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Beirlant, Y. Goegebeur, J. Teugels, and J. Segers, Statistics of Extremes, Wiley Series in Probability and Statistics, John Wiley & Sons, Chichester, UK, 2004, http://lstat.kuleuven.be/Wiley/. View at Publisher · View at Google Scholar · View at MathSciNet
  13. H. Drees, L. de Haan, and S. Resnick, “How to make a Hill plot,” The Annals of Statistics, vol. 28, no. 1, pp. 254–274, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. L. M. Dekkers and L. de Haan, “On the estimation of the extreme-value index and large quantile estimation,” The Annals of Statistics, vol. 17, no. 4, pp. 1795–1832, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Csörgö and L. Viharos, “Asymptotic normality of least-squares estimators of tail indices,” Bernoulli, vol. 3, no. 3, pp. 351–370, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  16. L. de Haan and S. Resnick, “On asymptotic normality of the Hill estimator,” Stochastic Models, vol. 14, no. 4, pp. 849–866, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  17. E. Haeusler and J. L. Teugels, “On asymptotic normality of Hill's estimator for the exponent of regular variation,” The Annals of Statistics, vol. 13, no. 2, pp. 743–756, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. I. Gomes and M. J. Martins, “Generalizations of the Hill estimator-asymptotic versus finite sample behaviour,” Journal of Statistical Planning and Inference, vol. 93, no. 1-2, pp. 161–180, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Resnick and C. Stǎricǎ, “Smoothing the Hill estimator,” Advances in Applied Probability, vol. 29, no. 1, pp. 271–293, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Csörgö, P. Deheuvels, and D. Mason, “Kernel estimates of the tail index of a distribution,” The Annals of Statistics, vol. 13, no. 3, pp. 1050–1077, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  21. J. Beirlant and A. Guillou, “Pareto index estimation under moderate right censoring,” Scandinavian Actuarial Journal, no. 2, pp. 111–125, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Beirlant, A. Guillou, G. Dierckx, and A. Fils-Villetard, “Estimation of the extreme value index and extreme quantiles under random censoring,” Extremes, vol. 10, no. 3, pp. 151–174, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. Caeiro, M. I. Gomes, and D. Pestana, “Direct reduction of bias of the classical Hill estimator,” REVSTAT Statistical Journal, vol. 3, no. 2, pp. 113–136, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M. I. Gomes, D. Pestana, and F. Caeiro, “A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator,” Statistics & Probability Letters, vol. 79, no. 3, pp. 295–303, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. I. Fraga Alves, “A location invariant Hill-type estimator,” Extremes, vol. 4, no. 3, pp. 199–217, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. Segers, “Abelian and Tauberian theorems on the bias of the Hill estimator,” Scandinavian Journal of Statistics, vol. 29, no. 3, pp. 461–483, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. P. Embrechts, C. Klüppelberg, and T. Mikosch, Modelling Extremal Events, vol. 33 of Applications of Mathematics (New York), Springer, Berlin, Germany, 1997. View at MathSciNet