Advertisement

Journal of Statistical Theory and Practice

, Volume 9, Issue 3, pp 571–599 | Cite as

Estimation of a Scale Second-Order Parameter Related to the PORT Methodology

  • Lígia Henriques-Rodrigues
  • M. Ivette Gomes
  • B. G. Manjunath
Article
First Online:
Received:
Accepted:

Abstract

For heavy right tails and under a semiparametric framework, we introduce a class of location-invariant estimators of a scale second-order parameter and study its asymptotic nondegenerate behavior. This class is based on the PORT methodology, with PORT standing for peaks over random thresholds. The consistency and asymptotic normality of the new class of estimators is achieved under a third-order condition on the right tail of the underlying model F for intermediate and large ranks, respectively. An illustration of the finite sample behavior of the estimators is provided through a Monte Carlo simulation study.

Keywords

Location/scale invariant estimation PORT methodology Semiparametric estimation Scale second-order parameters Statistics of extremes 

AMS 2000 Subject Classification

Primary 62G32, 62E20 Secondary 65C05 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Araújo Santos, P., M. I. Fraga Alves, and M. I. Gomes. 2006. Peaks over random threshold methodology for tail index and quantile estimation. Revstat 4(3), 227–247.MathSciNetzbMATHGoogle Scholar
  2. Beirlant, J., F. Caeiro, and M. I. Gomes. 2012. An overview and open research topics in the field of statistics of univariate extremes. Revstat 10(1), 1–31.MathSciNetzbMATHGoogle Scholar
  3. Caeiro, F., and M. I. Gomes. 2006. A new class of estimators of a “scale” second order parameter. Extremes 9, 193–211.MathSciNetCrossRefGoogle Scholar
  4. Davison, A. 1984. Modeling excesses over high threshold with an application. In Statistical extremes and applications, ed. J. Tiago de Oliveira, 461–482. Dordrecht, The Netherlands: D. Reidel.CrossRefGoogle Scholar
  5. Dekkers, A., J. Einmahl, and L. de Haan. 1989. A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17, 1833–1855.MathSciNetCrossRefGoogle Scholar
  6. Drees, H., A. Ferreira, and L. de Haan. 2004. On maximum likelihood estimation of the extreme value index. Ann. Appl. Prob., 14, 1179–1201.MathSciNetCrossRefGoogle Scholar
  7. Fraga Alves, M. I., M. I. Gomes, and L. de Haan. 2003. A new class of semi-parametric estimators of the second order parameter. Port. Math. 60(2), 194–213.MathSciNetzbMATHGoogle Scholar
  8. Fraga Alves, M. I., L. de Haan, and T. Lin. 2006. Third order extended regular variation. Pub. Inst. Math., 80(94), 109–120.MathSciNetCrossRefGoogle Scholar
  9. Geluk, J., and L. de Haan. 1987. Regular variation, extensions and Tauberian theorems. CWI Tract 40. Center for Mathematics and Computer Science, Amsterdam, The Netherlands.zbMATHGoogle Scholar
  10. Gnedenko, B. V. 1943. Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math., 44, 423–453.MathSciNetCrossRefGoogle Scholar
  11. Gomes, M. I. 2003. Stochastic processes in telecommunication traffic. In Mathematical techniques and problems in telecommunications, CIM edition, ed. C. Fernandes et al., 7–32. Coimbra, Portugal: CIM.Google Scholar
  12. Gomes, M. I., and A. Guillou. 2014. Extreme value theory and statistics of univariate extremes: A review. Int. Stat. Rev., in press. doi:10.1111/insr.12058.CrossRefMathSciNetGoogle Scholar
  13. Gomes, M. I., and L. Henriques-Rodrigues. 2012. Adaptive PORT-MVRB estimation of the extreme value index. In Studies in theoretical and applied statistics: Subseries B: Recent developments in modeling and applications in statistics, ed. P. Oliveira, M. G. Temido, C. Henriques, and M. Vichi, 117–125. Berlin, Heidelberg: Springer.Google Scholar
  14. Gomes, M. I., and M. J. Martins. 2002. “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. J. Stat. Plann. Inference, 93, 161–180.CrossRefGoogle Scholar
  15. Gomes, M. I., L. de Haan, and L. Peng. 2002. Semi-parametric estimation of the second order parameter—Asymptotic and finite sample behaviour. Extremes 5(4), 387–414.MathSciNetCrossRefGoogle Scholar
  16. Gomes, M. I., M. I. Fraga Alves, and P. Araújo Santos. 2008. PORT Hill and moment estimators for heavy-tailed models. Commun. Stat. Simul. Comput., 37, 1281–1306.MathSciNetCrossRefGoogle Scholar
  17. Gomes, M. I., F. Figueiredo, and M. M. Neves. 2012. Adaptive estimation of heavy right tails: Resampling-based methods in action. Extremes 15, 463–489.MathSciNetCrossRefGoogle Scholar
  18. Gomes, M. I., L. Henriques-Rodrigues, M. I. Fraga Alves, and B. G. Manjunath. 2013. Adaptive PORT-MVRB estimation: An empirical comparison of two heuristic algorithms. J. Stat. Comput. Simul. 83(6), 1129–1144.MathSciNetCrossRefGoogle Scholar
  19. de Haan, L. 1984. Slow variation and characterization of domains of attraction. In Statistical extremes and applications, ed. J. Tiago de Oliveira, 31–48. Dordrecht, The Netherlands: D. Reidel.CrossRefGoogle Scholar
  20. Hall, P. 1982. On estimating the endpoint of a distribution. Ann. Stat. 10, 556–568.MathSciNetCrossRefGoogle Scholar
  21. Hall, P., and A. W. Welsh. 1985. Adaptive estimates of parameters of regular variation. Ann. Stat., 13, 331–341.MathSciNetCrossRefGoogle Scholar
  22. Henriques-Rodrigues, L., and M. I. Gomes. 2009. High quantile estimation and the PORT methodology. Revstat 7(3), 245–264.MathSciNetzbMATHGoogle Scholar
  23. Henriques-Rodrigues, L., M. I. Gomes, M. I. Fraga Alves, and C. Neves. 2014. PORT-estimation of a shape second-order parameter. Revstat 12(3), 293–321.MathSciNetzbMATHGoogle Scholar
  24. Hüsler, J. 2009. Extreme value analysis in biometrics. Biometrical J. 51(2), 252–272.MathSciNetCrossRefGoogle Scholar
  25. Pickands, J. III. 1975. Statistical inference using extreme order statistics. Ann. Stat., 3, 119–131.MathSciNetCrossRefGoogle Scholar
  26. Reiss, R.-D., and M. Thomas. 2001. Statistical analysis of extreme values, with application to insurance, finance, hydrology and other fields, 2nd ed. Basel, Switzerland: Birkhäuser Verlag.zbMATHGoogle Scholar
  27. Reiss, R.-D., and M. Thomas. 2007. Statistical analysis of extreme values, with application to insurance, finance, hydrology and other fields, 3rd ed. Basel, Switzerland: Birkhäuser Verlag.zbMATHGoogle Scholar
  28. Resnick, S. I. 1997. Heavy tail modelling and teletraffic data. Ann. Stat. 25, 1805–1869.CrossRefGoogle Scholar
  29. van der Vaart, A. W. 1998. Asymptotic statistics. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  30. Wang, X., and S. Cheng. 2005. General regular variation of n-th order and the 2nd order edgeworth expansion of the extreme value distribution. Acta Math. Sin. 21(5), 1121–1130.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  • Lígia Henriques-Rodrigues
    • 1
    • 2
  • M. Ivette Gomes
    • 1
  • B. G. Manjunath
    • 1
  1. 1.Centro de Estatística e Aplicações and Departamento de Estatística e Investigação Operacional, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal
  2. 2.Instituto Politécnico de TomarTomarPortugal

Personalised recommendations

Cite article

Buy options