An efficient semiparametric maxima estimator of the extremal index

Extremes volume 18, pages585–603(2015)Cite this article

Abstract

The extremal index θ, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate θ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about θ with a standard extreme value analysis of block maxima to estimate marginal quantiles.

This is a preview of subscription content, log in to check access.

References

  1. Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of methods for estimating the extremal index. Extremes 3(1), 5–38 (2000)

    MATH  MathSciNet  Article  Google Scholar 

  2. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Chichester (2004)

    Google Scholar 

  3. Canty, A., Ripley, B.D.: boot: Bootstrap R (S-Plus) functions, R package version 1. 3–11 (2014)

  4. Chandler, R.E., Bate, S.B.: Inference for clustered data using the independence loglikelihood. Biometrika 94(1), 167–183 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  5. Chavez-Demoulin, V., Davison, A.C.: Modelling time series extremes. REVSTAT - Stat. J. 10(1), 109–133 (2012)

    MATH  MathSciNet  Google Scholar 

  6. Coles, S.G.: Modelling extreme multivariate events. PhD thesis, University of Sheffield, Sheffield, U.K. (1991)

  7. Dabrowska, D.M., Doksum, K.A., Miura, R.: Rank estimates in a class of semiparametric two-sample models. Ann. Inst. Statist. Math. 41(1), 63–79 (1989)

    MATH  MathSciNet  Google Scholar 

  8. Davis, R., Resnick, S.: Basic properties and prediction of max-ARMA processes. Adv. Appl. Prob. 21, 781–803 (1989)

    MATH  MathSciNet  Article  Google Scholar 

  9. Davison, A.C., Hinkley, D.V.: Bootstrap Methods and Their Application. Cambridge University Press, New York (1997)

    Google Scholar 

  10. Deheuvels, P.: Point processes and multivariate extreme values. J. Multivariate. Anal. 13(2), 257–272 (1983)

    MATH  MathSciNet  Article  Google Scholar 

  11. Fawcett, L., Walshaw, D.: Estimating return levels from serially dependent extremes. Environmetrics 23(3), 272–283 (2012)

    MathSciNet  Article  Google Scholar 

  12. Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Statist. Soc. B 65(2), 545–556 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  13. Ferro, C.A.T., Pezzulli, S.: Rao-Blackwellised estimators for parameters of extreme-value models http://empslocal.ex.ac.uk/people/staff/ferro/Publications/raoblackwell.pdf (2005)

  14. Gomes, M.I.: On the estimation of parameters of rare events in environmental time series. In: Barnett, V, Turkman, K (eds.) Statistics for the Environment 2: Water Related Issues, pp. 225–241. Wiley, New York (1993)

    Google Scholar 

  15. Hayfield, T., Racine, J.S.: Nonparametric econometrics: The np package. J. Stat. Softw. 27(5), 1–32 (2008)

    Article  Google Scholar 

  16. Laurini, F., Tawn, J.A.: New estimators for the extremal index and other cluster characteristics. Extremes 6, 189–211 (2003)

    MATH  MathSciNet  Article  Google Scholar 

  17. Leadbetter, M., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and series. Springer Verlag, New York (1983)

    Google Scholar 

  18. Northrop, P.J.: Semiparametric estimation of the extremal index using block maxima. Tech. Rep. 259, University College London (2005)

  19. Patton, A., Politis, D., White, H.: Correction to Automatic block-length selection for the dependent bootstrap by D. Politis and H. White. Econ. Rev. 28(4), 372–375 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  20. Politis, D., Romano, J.: The stationary bootstrap. J. Am. Statist. Ass. 89(428), 1303–1313 (1994)

    MATH  MathSciNet  Article  Google Scholar 

  21. Prescott, P., Walden, A.: Maximum likelihood estimation of the parameters of the generalized extreme value distribution. Biometrika 67(3), 723–724 (1980)

    MathSciNet  Article  Google Scholar 

  22. Robert, C.Y.: Inference for the limiting cluster size distribution of extreme values. Ann. Statist. 37(1), 271–310 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  23. Robert, C.Y.: Automatic declustering of rare events. Biometrika 100(3), 587–606 (2013)

    MATH  MathSciNet  Article  Google Scholar 

  24. Robert, C.Y., Segers, J., Ferro, C.A.T.: A sliding blocks estimator for the extremal index. Electron. J. Stat. 3, 993–1020 (2009)

    MATH  MathSciNet  Article  Google Scholar 

  25. Smith, R.L.: Maximum likelihood estimation in a class of non-regular cases. Biometrika 72(1), 67–92 (1985)

    MATH  MathSciNet  Article  Google Scholar 

  26. Smith, R.L.: A theoretical comparison of the annual maximum and threshold approaches to extreme value analysis. Tech. Rep. 53, University of Surrey (1987)

  27. Smith, R.L.: The extremal index for a Markov Chain. J. App. Prob. 29(1), 37–45 (1992)

    MATH  Article  Google Scholar 

  28. Smith, R.L., Weissman, I.: Estimating the extremal index. J. R. Statist. Soc. B. 56(3), 515–528 (1994)

    MATH  MathSciNet  Google Scholar 

  29. Süveges, M.: Likelihood estimation of the extremal index. Extremes 10, 41–55 (2007)

    MATH  MathSciNet  Article  Google Scholar 

  30. Süveges, M., Davison, A.C.: Model misspecification in peaks over threshold analysis. Ann. Appl. Statist. 4(1), 203–221 (2010)

    MATH  Article  Google Scholar 

  31. White, H.: Maximum likelihood estimation of misspecified models. Econometrika 50(1), 1–25 (1982)

    MATH  Article  Google Scholar 

Download references

Author information

Affiliations

  1. Department of Statistical Science, University College London, Gower Street, London, WC1E 6BT, UK

    Paul J. Northrop

Authors
  1. Paul J. Northrop
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul J. Northrop.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Northrop, P.J. An efficient semiparametric maxima estimator of the extremal index. Extremes 18, 585–603 (2015). https://doi.org/10.1007/s10687-015-0221-5

Download citation

Keywords

  • Block maxima
  • Extremal index
  • Extreme value theory
  • Sea-surge heights
  • Semiparametric estimation

AMS 2000 Subject Classifications

  • 62G32
  • 62F03
  • 62F05
  • 62P12
  • 62P35