Extremes

, 11:217

Bayesian inference for clustered extremes

Article

Abstract

We consider Bayesian inference for the extremes of dependent stationary series. We discuss the virtues of the Bayesian approach to inference for the extremal index, and for related characteristics of clustering behaviour. We develop an inference procedure based on an automatic declustering scheme, and using simulated data we implement and assess this procedure, making inferences for the extremal index, and for two cluster functionals. We then apply our procedure to a set of real data, specifically a time series of wind-speed measurements, where the clusters correspond to storms. Here the two cluster functionals selected previously correspond to the mean storm length and the mean inter-storm interval. We also consider inference for long-period return levels, advocating the posterior predictive distribution as being most representative of the information required by engineers interested in design level specifications.

Keywords

Bayesian inference Extremal index MCMC Clusters Extreme values 

AMS 2000 Subject Classification

62G32 

References

  1. Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of methods for estimating the extremal index. Extremes 3, 5–38 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. Bottolo, P., Consonni, G., Dellaportos, P., Lijoi, A.: Bayesian analysis of extreme values by mixture modelling. Extremes 6, 25–48 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. British Standards Institution: Code of Basic Data for the Design of Buildings; CP 3, ch. V, Loading; part 2, Wind Loads. British Standards Institution, London. Waves in a record. Proc. R. Soc. Lond. A 247, 22–48 (1997)Google Scholar
  4. Coles, S.G.: An introduction to statistical modeling of extreme values. Springer, London (2001)MATHGoogle Scholar
  5. Coles, S.G., Powell, E.: Bayesian methods in extreme value modelling: a review and new developments. Internat. Statist. Review. 64, 119–136 (1996)MATHCrossRefGoogle Scholar
  6. Coles, S.G., Tawn, J.A.: A bayesian analysis of extreme rainfall data. Appl. Statist. 45, 463–478 (1996)CrossRefGoogle Scholar
  7. Fawcett, L.: Statistical methodology for the estimation of environmental extremes. Ph.D. thesis, University of Newcastle upon Tyne (2005)Google Scholar
  8. Fawcett, L., Walshaw, D.: A hierarchical model for extreme wind speeds. Appl. Stat. 55(5), 631–646 (2006a)MATHMathSciNetGoogle Scholar
  9. Fawcett, L., Walshaw, D.: Markov chain models for extreme wind speeds. Environmetrics 17(8), 795–809 (2006b)CrossRefMathSciNetGoogle Scholar
  10. Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18(2), 173–188 (2007)CrossRefMathSciNetGoogle Scholar
  11. Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Statist. Soc. B 65, 545–556 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. Gomes, M.I.: On the estimation of parameters of rare events in environmental time series. In: Barnett, V., Turkman K.F. (eds.) Statistics for the Environment 2: Water Related Issues, pp. 225–241 (1993)Google Scholar
  13. Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Prob. Theory Rel. Fields 78, 97–112 (1988)MATHCrossRefGoogle Scholar
  14. Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Series. Springer-Verlag, New York (1983)Google Scholar
  15. Leadbetter, M.R., Rootzén, H.: Extremal theory for stochastic processes. Ann. Probab. 16, 431–476 (1988)MATHCrossRefMathSciNetGoogle Scholar
  16. Loynes, R.M.: Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999 (1965)MATHCrossRefMathSciNetGoogle Scholar
  17. Newell, G.F.: Asymptotic extremes for m-dependent random variables. Ann. Math Statist. 35, 1322–1325 (1964)MATHCrossRefMathSciNetGoogle Scholar
  18. O’Brien, G.L.: The maximum term of uniformly mixing stationary processes. Z. Wahrscheinlichkeitsth. 30, 57–63 (1974)MATHCrossRefMathSciNetGoogle Scholar
  19. Smith, R.L.: The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45 (1992)MATHCrossRefGoogle Scholar
  20. Smith, R.L.: Bayesian and frequentist approaches to parametric predictive inference (with discussion). In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 6, pp. 589–612. Oxford University Press (1999)Google Scholar
  21. Smith, R.L., Goodman, D.J.: Bayesian risk analysis. In: Embrechts, P. (ed.) Extremes and Integrated Risk Management, pp. 235–251. Risk Books: London (2000)Google Scholar
  22. Smith, A.F.M., Roberts, G.O.: Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Statist. Soc., B 55, 3–23 (1993)MATHMathSciNetGoogle Scholar
  23. Smith, E.L., Walshaw, D.: Modelling bivariate extremes in a region. In: Bernardo, J.M., Bayarri, M.J., Berger, J.O., Dawid, A.P., Heckerman, D., Smith, A.F.M., West, M. (eds.) Bayesian Statistics, vol. 7, pp. 681–690. Oxford University Press (2003)Google Scholar
  24. Smith, R.L., Weissman, I.: Estimating the extremal index. J. R. Statist. Soc., B 56, 515–528 (1994)MATHMathSciNetGoogle Scholar
  25. Smith, R.L., Coles, S.G., Tawn, J.A.: Markov chain models for threshold exceedances. Biometrika 84, 249–268 (1997)MATHCrossRefMathSciNetGoogle Scholar
  26. Stephenson, A., Tawn, J.A.: Bayesian inference for extremes: accounting for the three extremal types. Extremes 7, 291–307 (2004)MATHCrossRefMathSciNetGoogle Scholar
  27. Tawn, J.A.: Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988)MATHCrossRefMathSciNetGoogle Scholar
  28. Venzon, D.J., Moolgavkar, S.H.: Profile-likelihood-based confidence intervals. Appl. Stat. 37, 87–94 (1988)CrossRefGoogle Scholar
  29. Walshaw, D.: Getting the most from your extreme wind data: a step by step guide. J. Res. Natl. Inst. Stand. Technol. 99, 399–411 (1994)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNewcastle UniversityNewcastleUK

Personalised recommendations