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Journal of Statistical Theory and Practice

, Volume 9, Issue 1, pp 184–199 | Cite as

Modeling Extreme Events: Sample Fraction Adaptive Choice in Parameter Estimation

  • M. Manuela Neves
  • M. Ivette Gomes
  • Fernanda Figueiredo
  • Dora Prata Gomes
Article

Abstract

When modeling extreme events, there are a few primordial parameters, among which we refer to the extreme value index (EVI) and the extremal index (EI). Under a framework related to large values, the EVI measures the right tail weight of the underlying distribution and the EI characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semiparametric estimators of these parameters show the same type of behavior: nice asymptotic properties but a high variance for small values of k, the number of upper order statistics used in the estimation, and a high bias for large values of k. This brings a real need for the choice of k. Choosing some well-known estimators of those two parameters, we revisit the application of a heuristic algorithm for the adaptive choice of k. A simulation study illustrates the performance of the proposed algorithm.

Keywords

Adaptive choice Extremal index Extreme value index Sample fraction Semiparametric estimation 

AMS Subject Classification

Primary 62G32 Secondary 65C05 

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References

  1. Alpuim, M. T. 1989. An extremal markovian sequence. J. Appl. Prob., 26, 219–222.MathSciNetCrossRefGoogle Scholar
  2. Beirlant, J., Y. Goegebeur, J. Segers, J. Teugels, D. Waal, and C. Ferro. 2004. Statistics of extremes: Theory and applications. New York, NY: John Wiley & Sons.CrossRefGoogle Scholar
  3. Bingham, N. H., C. M. Goldie, and J. L. Teugels. 1987. Regular variation. New York, NY: Cambridge University Press.CrossRefGoogle Scholar
  4. Caeiro, F., M. I. Gomes, and D. D. Pestana. 2005. Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 113–136.MathSciNetMATHGoogle Scholar
  5. Caeiro, F., M. I. Gomes, and L. Henriques-Rodrigues. 2009. Reduced-bias tail index estimators under a third order framework. Commun. Stat. Theory Methods, 38(7), 1019–1040.MathSciNetCrossRefGoogle Scholar
  6. Davidson, A. 2011. Statistics of extremes. Courses 2011–2012. École Polytechnique Fédérale de Lausanne EPFL, Lausanne, Switzerland.Google 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.MathSciNetMATHGoogle Scholar
  8. Gomes, M. I. 1990. Statistical inference in an extremal Markovian model. In COMPSTAT 1990: Proceedings in Computational Statistics, eds. K. Momirovic and V. Mildner, 257–262. Heidelberg: Physica-Verlag.Google Scholar
  9. Gomes, M. I. 1993a. Modelos extremais em esquemas de dependência. In Estatística Robusta, Extremos e Mais Alguns Temas, ed. D. Pestana, 209–220. Lisboa: Salamandra.Google Scholar
  10. Gomes, M. I. 1993b. On the estimation of parameters of rare events in environmental time series. In Statistics for the environment, eds. V. Barnett and K. F. Turkman, 226–241. New York, NY: John Wiley & Sons.Google Scholar
  11. Gomes, M. I., and M. J. Martins. 2002. “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5–31.MathSciNetCrossRefGoogle Scholar
  12. Gomes, M. I., L. de Haan, and L. Henriques-Rodrigues. 2008a. Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. R. Stat. Soc., B70(1), 31–52.MathSciNetMATHGoogle Scholar
  13. Gomes, M. I., A. Hall, and C. Miranda. 2008b. Subsampling techniques and the jackknife methodology in the estimation of the extremal index. J. Comput. Stat. Data Anal., 52(4), 2022–2041.MathSciNetCrossRefGoogle Scholar
  14. 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
  15. Gomes, M. I., M. J. Martins, and M. M. Neves. 2013. Generalised jackknife-based estimators for univariate extreme-value modelling. Commun. Stat. Theory Methods, 42(7), 1227–1245.CrossRefGoogle Scholar
  16. Gray, H. L., and W. R. Schucany. 1972. The generalized jackknife statistic. New York, NY: Marcel Dekker.MATHGoogle Scholar
  17. Hill, B. 1975. A simple general approach to inference about the tail of a distribution. Ann. Stat., 3, 1163–1174.MathSciNetCrossRefGoogle Scholar
  18. Hsing, T., J. Husler, and M. R. Leadbetter. 1988. On exceedance point process for a stationary sequence. Probab. Theory Related Fields, 78, 97–112.MathSciNetCrossRefGoogle Scholar
  19. Leadbetter, M. R. 1983. Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete, 65(2), 291–306.MathSciNetCrossRefGoogle Scholar
  20. Leadbetter, M. R., G. Lindgren, and H. Rootzén. 1983. Extremes and related properties of random sequences and series. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
  21. Leadbetter, M. R., and L. Nandagopalan. 1989. On exceedance point process for stationary sequences under mild oscillation restrictions. In Extreme value theory: Proceedings, Oberwolfach 1987, ed. J. Hüsler and R. D. Reiss, Lecture Notes in Statistics 52, 69–80. Berlin, Germany: Springer-Verlag.CrossRefGoogle Scholar
  22. Martins, A. P., and H. Ferreira. 2004. The extremal index of sub-sampled processes. J. Stat. Plan. Inference, 124, 145–152.MathSciNetCrossRefGoogle Scholar
  23. Nandagopalan, S. 1990. Multivariate extremes and estimation of the extremal index. PhD thesis, University of North Carolina, Chapel Hill, NC.MATHGoogle Scholar
  24. Nandagopalan, S., and H. Rootzén. 1988. Extremal theory for stochastic processes. Ann. Probab., 16(2), 431–478.MathSciNetCrossRefGoogle Scholar
  25. O’Brien, G. 1987. Extreme values for stationary and Markov sequences. Ann. Probab., 15(1), 281–289.MathSciNetCrossRefGoogle Scholar
  26. Prata Gomes, D., and M. M. Neves. 2011. Resampling methodologies and the estimation of parameters of rare events. In Numerical analysis and applied mathematics (ICNAAM 2011), AIP Conf. Proc., 1389, 1475–1478.Google Scholar
  27. Robinson, M. E., and J. A. Tawn. 2000. Extremal analysis of processes sampled at different frequencies. J. R. Stat. Soc. B, 62, 117–135.MathSciNetCrossRefGoogle Scholar
  28. Scotto, M., K. F. Turkman, and C. W. Anderson. 2003. Extremes of some sub-sampled time series. J. Time Series Anal., 24(5), 505–512.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  • M. Manuela Neves
    • 1
  • M. Ivette Gomes
    • 2
  • Fernanda Figueiredo
    • 3
  • Dora Prata Gomes
    • 4
  1. 1.ISA and CEAULUniversidade de LisboaLisboaPortugal
  2. 2.FCUL, DEIO, and CEAULUniversidade de LisboaLisboaPortugal
  3. 3.FEP and CEAULUniversidade do PortoPortoPortugal
  4. 4.Mathematics Department, FCT and CMAUniversidade Nova de LisboaLisboa, CaparicaPortugal

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