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.
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References
Ancona-Navarrete, M.A., Tawn, J.A.: A comparison of methods for estimating the extremal index. Extremes 3, 5–38 (2000)
Bottolo, P., Consonni, G., Dellaportos, P., Lijoi, A.: Bayesian analysis of extreme values by mixture modelling. Extremes 6, 25–48 (2003)
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)
Coles, S.G.: An introduction to statistical modeling of extreme values. Springer, London (2001)
Coles, S.G., Powell, E.: Bayesian methods in extreme value modelling: a review and new developments. Internat. Statist. Review. 64, 119–136 (1996)
Coles, S.G., Tawn, J.A.: A bayesian analysis of extreme rainfall data. Appl. Statist. 45, 463–478 (1996)
Fawcett, L.: Statistical methodology for the estimation of environmental extremes. Ph.D. thesis, University of Newcastle upon Tyne (2005)
Fawcett, L., Walshaw, D.: A hierarchical model for extreme wind speeds. Appl. Stat. 55(5), 631–646 (2006a)
Fawcett, L., Walshaw, D.: Markov chain models for extreme wind speeds. Environmetrics 17(8), 795–809 (2006b)
Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18(2), 173–188 (2007)
Ferro, C.A.T., Segers, J.: Inference for clusters of extreme values. J. R. Statist. Soc. B 65, 545–556 (2003)
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)
Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Prob. Theory Rel. Fields 78, 97–112 (1988)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Series. Springer-Verlag, New York (1983)
Leadbetter, M.R., Rootzén, H.: Extremal theory for stochastic processes. Ann. Probab. 16, 431–476 (1988)
Loynes, R.M.: Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999 (1965)
Newell, G.F.: Asymptotic extremes for m-dependent random variables. Ann. Math Statist. 35, 1322–1325 (1964)
O’Brien, G.L.: The maximum term of uniformly mixing stationary processes. Z. Wahrscheinlichkeitsth. 30, 57–63 (1974)
Smith, R.L.: The extremal index for a Markov chain. J. Appl. Prob. 29, 37–45 (1992)
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)
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)
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)
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)
Smith, R.L., Weissman, I.: Estimating the extremal index. J. R. Statist. Soc., B 56, 515–528 (1994)
Smith, R.L., Coles, S.G., Tawn, J.A.: Markov chain models for threshold exceedances. Biometrika 84, 249–268 (1997)
Stephenson, A., Tawn, J.A.: Bayesian inference for extremes: accounting for the three extremal types. Extremes 7, 291–307 (2004)
Tawn, J.A.: Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988)
Venzon, D.J., Moolgavkar, S.H.: Profile-likelihood-based confidence intervals. Appl. Stat. 37, 87–94 (1988)
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)
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Fawcett, L., Walshaw, D. Bayesian inference for clustered extremes. Extremes 11, 217–233 (2008). https://doi.org/10.1007/s10687-007-0054-y
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DOI: https://doi.org/10.1007/s10687-007-0054-y