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A Point Process Characterization of Extremes

  • Stuart Coles
Part of the Springer Series in Statistics book series (SSS)

Abstract

There are different ways of characterizing the extreme value behavior of a process, and a particularly elegant formulation is derived from the theory of point processes. The mathematics required for a formal treatment of this theory is outside the scope of this book, but we can again give a more informal development. This requires just basic ideas from point process theory. In a sense, the point process characterization leads to nothing new in terms of statistical models; all inferences made using the point process methodology could equally be obtained using an appropriate model from earlier chapters. However, there are two good reasons for considering this approach. First, it provides an interpretation of extreme value behavior that unifies all the models introduced so far; second, the model leads directly to a likelihood that enables a more natural formulation of non-stationarity in threshold excesses than was obtained from the generalized Pareto model discussed in Chapters 4 and 6.

Keywords

Poisson Process Point Process Return Level Homogeneous Poisson Process Threshold Exceedance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Further Reading

  1. Cox, D. R. and Isham, V. (1980). Point Processes. Chapman and Hall, London.Google Scholar
  2. Pickands, J. (1971). The two-dimensional poisson process and extremal processes. Journal of Applied Probability 8, 745–756.MathSciNetCrossRefMATHGoogle Scholar
  3. Leadbetter, M. R. (1983). Extremes and local dependence in stationary-sequences. Zeit. Wahrscheinl.-theorie 65,291–306.Google Scholar
  4. Leadbetter, M. R., Lindgren, G., and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Series. Springer Verlag, New York.Google Scholar
  5. Smith, R. L. (1986). Extreme value theory based on the r largest annual events. Journal of Hydrology 86, 27–43.CrossRefGoogle Scholar
  6. Smith, R. L. (1989a). Extreme value analysis of environmental time series: an example based on ozone data (with discussion). Statistical Science 4, 367–393.MathSciNetCrossRefMATHGoogle Scholar
  7. Coles, S. G. and Tawn, J. A. (1996a). A Bayesian analysis of extreme rainfall data. Applied Statistics 45, 463–478.CrossRefGoogle Scholar
  8. Coles, S. G. and Tawn, J. A. (1996b). Modelling extremes of the areal rainfall process. Journal of the Royal Statistical Society, B 58, 329–347.Google Scholar
  9. Coles, S. G., Tawn, J. A., and Smith, R. L. (1994). A seasonal Markov model for extremely low temperatures. Environmetrics 5, 221–239. generalized Pareto distribution model for high concentrations in short-range atmospheric dispersion. Environmetrics 6, 595–606. Google Scholar
  10. Moore, R. J. (1987). Combined regional flood frequency analysis and regression on catchment characteristics by maximum likelihood estimation. In Singh, V. P., editor, Regional Flood Frequency Analysis,pages 119–131. Reidel, Dordrecht.Google Scholar
  11. Coles, S. G. and Tawn, J. A. (1991). Modelling extreme multivariate events. Journal of the Royal Statistical Society, B 53, 377–392.Google Scholar
  12. Morton, I. D., Bowers, J., and Mould, G. (1997). Estimating return period wave heights and wind speeds using a seasonal point process model. Coastal Engineering 31, 305–326.Google Scholar

Copyright information

© Springer-Verlag London 2001

Authors and Affiliations

  • Stuart Coles
    • 1
  1. 1.Department of MathematicsUniversity of BristolBristolUK

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