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Bayesian Inference for Extremes: Accounting for the Three Extremal Types

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The Extremal Types Theorem identifies three distinct types of extremal behaviour. Two different strategies for statistical inference for extreme values have been developed to exploit this asymptotic representation. One strategy uses a model for which the three types are combined into a unified parametric family with the shape parameter of the family determining the type: positive (Fréchet), zero (Gumbel), and negative (negative Weibull). This form of approach never selects the Gumbel type as that type is reduced to a single point in a continuous parameter space. The other strategy first selects the extremal type, based on hypothesis tests, and then estimates the best fitting model within the selected type. Such approaches ignore the uncertainty of the choice of extremal type on the subsequent inference. We overcome these deficiencies by applying the Bayesian inferential framework to an extended model which explicitly allocates a non-zero probability to the Gumbel type. Application of our procedure suggests that the effect of incorporating the knowledge of the Extremal Types Theorem into the inference for extreme values is to reduce uncertainty, with the degree of reduction depending on the shape parameter of the true extremal distribution and the prior weight given to the Gumbel type.

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  • Achcar, A.J., Bolfarine, H., and Pericchi, L.R., “Transformation of survival data to an extreme value distribution,” Statistician 36, 229–234, (1987).

    Google Scholar 

  • Ashour, S.K., and El-Adl, Y.M., “Bayesian estimation of the parameters of the extreme value distribution,” Egypt. Stat. J. 24, 140–152, (1980).

    Google Scholar 

  • Bottolo, P., Consonni, G., Dellaportas, P., and Lijoi, A., “Bayesian analysis of extreme values by mixture modelling,” Extremes 6, 25–48, (2003).

    Article  MathSciNet  Google Scholar 

  • Cohen, J.P., “Convergence rates for the ultimate and penultimate approximations in extreme-value theory,” Adv. Appl. Probab. 14, 833–854, (1982a).

    Google Scholar 

  • Cohen, J.P., “The penultimate form of approximation to normal extremes,” Adv. Appl. Probab. 14, 324–339, (1982b).

    Google Scholar 

  • Coles, S.G., An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London, 2001.

    Google Scholar 

  • Coles, S.G., and Powell, E.A., “Bayesian methods in extreme value modelling: a review and new developments,” Int. Stat. Rev. 64, 119–136, (1996).

    Google Scholar 

  • Coles, S.G., and Tawn, J.A., “Statistics of coastal flood prevention,” Philos. Trans. R. Soc. Lond., A 332, 457–476, (1990).

    Google Scholar 

  • Coles, S.G., and Tawn, J.A., “A Bayesian analysis of extreme rainfall data,” Appl. Stat. 45, 463–478, (1996).

    Google Scholar 

  • Davison, A.C., and Smith, R.L., “Models for exceedances over high thresholds,” J. R. Stat. Soc., B 52, 393–442, (1990).

    Google Scholar 

  • Dekkers, A.L.M., and de Haan, L., “On the estimation of the extreme-value index and large quantile estimation,” Ann. Stat. 17, 1795–1832, (1989).

    Google Scholar 

  • Dixon, M.J., and Tawn, J.A., “Trends in U.K. extreme sea levels: a spatial approach,” Geophys. J. Int. 111, 607–616, (1992).

    Google Scholar 

  • Engelund, S., and Rackwitz, R., “On predictive distribution functions for the three asymptotic extreme value distributions,” Struct. Saf. 11, 255–258, (1992).

    Article  Google Scholar 

  • Fisher, R.A., and Tippett, L.H.C., “Limiting forms of frequency distributions of the largest or smallest member of a sample,” Proc. Camb. Philos. Soc. 24, 180–190, (1928).

    Google Scholar 

  • Galambos, J., “An statistical test for extreme value distributions,” Tech. Rep. 32, Coll. Math. Soc. Janos. Bolyai, Budapest, 1980.

  • Geman, S., and Geman, D., “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741, (1984).

    Google Scholar 

  • Gomes, M.I., “A note on statistical choice of extremal models,” Tech. Rep., Act. IX Jorn. Matem. Hispano-Lusas, Salamanca, II, 1982.

  • Green, P.J., “Reversible jump MCMC computation and Bayesian model determination,” Biometrica 82, 711–732, (1995).

    Google Scholar 

  • Gumbel, E.J., Statistics of Extremes, Columbia University Press, New York, 1958.

    Google Scholar 

  • Hastings, W.K., “Monte Carlo sampling methods using markov chains and their applications,” Biometrica 57, 97–109, (1970).

    Google Scholar 

  • Hosking, J.R.M., “Testing whether the shape parameter is zero in the generalized extreme-value distribution,” Biometrika 71, 367–374, (1984).

    Google Scholar 

  • Jenkinson, A.F., “The frequency distribution of the annual maximum (or minimum) of meteorological elements,” Quant. J. R. Met. Soc. 81, 158–171, (1955).

    Google Scholar 

  • Leadbetter, M.R., Lindgren, G., and Rootzén, H., Extremes and Related Properties of Random Sequences and Series, Springer-Verlag, New York, 1983.

    Google Scholar 

  • Lingappaiah, G.S., “Bayesian prediction regions for the extreme order statistics,” Biom. J. 26, 49–56, (1984).

    Google Scholar 

  • Pickands, J., “The two-dimensional Poisson process and extremal processes,” J. Appl. Probab. 8, 745–756, (1971).

    Google Scholar 

  • Pickands, J., “Statistical inference using extreme order statistics,” Ann. Stat. 3, 119–131, (1975).

    Google Scholar 

  • Pickands, J., “Bayes quantile estimation and threshold selection for the generalized pareto family,” in: Extreme Value Theory and Applications (J. Galambos, S. Leigh, and E. Simiu, eds), Kluwer, Amsterdam, 123–138, (1994).

    Google Scholar 

  • Reiss, R.D., and Thomas, M., Statistical Analysis of Extreme Values, from Insurance, Finance Hydrology and Other Fields, Birkhauser, New York, 2001.

    Google Scholar 

  • Rootzén, H., and Tajvidi, N., “Extreme value statistics and wind storm losses: a case study,” J. Scand. Actuar. 1, 70–94, (1997).

    Google Scholar 

  • Smith, R.L., “Approximations in extreme value theory,” Tech. Rep. 205, Department of Statistics, University of North Carolina, Chapel Hill, 1987.

  • Smith, R.L., “Extreme value analysis of environmental time series: an example based on ozone data (with discussion),” Stat. Sci. 4, 367–393, (1989).

    Google Scholar 

  • Smith, R.L., “Bayesian and frequentist approaches to parametric predictive inference (with discussion),” in: Bayesian Statistics 6 (J.M. Bernardo, J.O. Berger, A.P. Dawid, and A.F.M. Smith, eds), Oxford University Press, 589–612, (1999).

  • Smith, R.L., and Goodman, D.J., “Bayesian risk analysis,” in: Extremes and Integrated Risk Management (P. Embrechts, ed) Risk Books, London, 235–251, (2000).

    Google Scholar 

  • Smith, R.L., and Naylor, J.C., “A comparison of maximum likelihood and Bayesian estimators for the three parameter Weilbull distribution,” Appl. Stat. 36, 358–369, (1987).

    Google Scholar 

  • Tawn, J.A., “Estimating probabilities of extreme sea-levels,” Appl. Stat. 41, 77–93, (1992).

    Google Scholar 

  • Tiago de Oliveira, J., and Gomes, M.I., “Two test statistics for choice of univariate extreme models,” in: Statistical Extremes and Applications (J. Tiago de Oloveira, ed) Reidel, Dordrecht, 651–668, (1984).

    Google Scholar 

  • van Montfort, M.A.J., “An asymmetric test on the type of distribution of extremes,” Tech. Rep. 73-18, Medelelugen Landbbouwhoge-School Wageningen, 1973.

  • van Montfort, M.A.J., and Otten, A., “On testing a shape parameter in the presence of a scale parameter,” Math. Oper.Forsch. Stat., Ser. Stat. 9, 91–104, (1978).

    Google Scholar 

  • Walshaw, D., “Modelling extreme wind speeds in regions prone to hurricanes,” Appl. Stat. 49, 51–62, (2000).

    Google Scholar 

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Correspondence to Alec Stephenson.

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Stephenson, A., Tawn, J. Bayesian Inference for Extremes: Accounting for the Three Extremal Types. Extremes 7, 291–307 (2004).

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