, Volume 20, Issue 2, pp 239–263 | Cite as

A Poisson process reparameterisation for Bayesian inference for extremes

  • Paul SharkeyEmail author
  • Jonathan A. Tawn
Open Access


A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more difficult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and efficient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK.


Poisson processes Extreme value theory Bayesian inference Reparameterisation Covariate modelling 

AMS 2000 Subject Classifications

60G70 62F15 62P12 62G32 


  1. Attalides, N.: Threshold-Based Extreme Value Modelling. PhD thesis, UCL (University College London) (2015)Google Scholar
  2. Chavez-Demoulin, V., Davison, A.C.: Generalized additive modelling of sample extremes. J. R. Stat. Soc.: Ser. C: Appl. Stat. 54(1), 207–222 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Coles, S.G.: An Introduction to Statistical Modeling of Extreme Values. Springer (2001)Google Scholar
  4. Coles, S.G., Tawn, J.A.: A Bayesian analysis of extreme rainfall data. Appl. Stat. 45, 463–478 (1996)CrossRefGoogle Scholar
  5. Cox, D.R., Reid, N.: Parameter orthogonality and approximate conditional inference (with discussion). J. R. Stat. Soc. Ser. B Methodol. 49(1), 1–39 (1987)zbMATHGoogle Scholar
  6. Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds (with discussion). J. R. Stat. Soc. Ser. B Methodol. 52(3), 393–442 (1990)zbMATHGoogle Scholar
  7. Efron, B., Hinkley, D.V.: Assessing the accuracy of the maximum likelihood estimator: Observed versus expected fisher information. Biometrika 65(3), 457–483 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Gander, W.: On Halley’s iteration method. Am. Math. Mon. 92(2), 131–134 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Hills, S.E., Smith, A.F.: Parameterization issues in Bayesian inference. Bayesian Statistics 4, 227–246 (1992)MathSciNetGoogle Scholar
  10. Hurrell, J.W., Kushnir, Y., Ottersen, G., Visbeck, M.: An overview of the North Atlantic oscillation. Geophysical Monograph-American Geophysical Union 134, 1–36 (2003)Google Scholar
  11. Northrop, P.J., Attalides, N.: Posterior propriety in Bayesian extreme value analyses using reference priors. Stat. Sin. 26(2), 721–743 (2016)MathSciNetzbMATHGoogle Scholar
  12. Northrop, P.J., Jonathan, P.: Threshold modelling of spatially dependent non-stationary extremes with application to hurricane-induced wave heights. Environmetrics 22(7), 799–809 (2011)MathSciNetCrossRefGoogle Scholar
  13. Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Robert, C., Casella, G.: Introducing Monte Carlo Methods with R. Springer Science & Business Media (2009)Google Scholar
  15. Roberts, G.O., Rosenthal, J.S., et al.: Optimal scaling for various Metropolis-Hastings algorithms. Stat. Sci. 16(4), 351–367 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Smith, R.L.: Maximum likelihood estimation in a class of nonregular cases. Biometrika 72(1), 67–90 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Smith, R.L.: Discussion of “Parameter orthogonality and approximate conditional inference” by D.R. Cox and N. Reid. J. R. Stat. Soc. Ser. B Methodol. 49(1), 21–22 (1987a)MathSciNetGoogle Scholar
  18. Smith, R.L.: A theoretical comparison of the annual maximum and threshold approaches to extreme value analysis. Technical Report 53, University of Surrey (1987b)Google Scholar
  19. Smith, R.L.: Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone. Stat. Sci. 4(4), 367–377 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Stephenson, A.: Bayesian inference for extreme value modelling. Extreme Value Modeling and Risk Analysis: Methods and Applications, pp. 257–280 (2016)Google Scholar
  21. Tawn, J.A.: Discussion of “Parameter orthogonality and approximate conditional inference” by D.R. Cox and N. Reid. J. R. Stat. Soc. Ser. B Methodol. 49(1), 33–34 (1987)MathSciNetGoogle Scholar
  22. Wadsworth, J.L., Tawn, J.A., Jonathan, P.: Accounting for choice of measurement scale in extreme value modeling. Ann. Appl. Stat. 4(3), 1558–1578 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.STOR-i Centre for Doctoral Training, Department of Mathematics and StatisticsLancaster UniversityLancasterUK

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