Electronic Journal of Statistics

Bayesian inference for multivariate extreme value distributions

Clément Dombry, Sebastian Engelke, and Marco Oesting

Full-text: Open access


Statistical modeling of multivariate and spatial extreme events has attracted broad attention in various areas of science. Max-stable distributions and processes are the natural class of models for this purpose, and many parametric families have been developed and successfully applied. Due to complicated likelihoods, the efficient statistical inference is still an active area of research, and usually composite likelihood methods based on bivariate densities only are used. Thibaud et al. (2016) use a Bayesian approach to fit a Brown–Resnick process to extreme temperatures. In this paper, we extend this idea to a methodology that is applicable to general max-stable distributions and that uses full likelihoods. We further provide simple conditions for the asymptotic normality of the median of the posterior distribution and verify them for the commonly used models in multivariate and spatial extreme value statistics. A simulation study shows that this point estimator is considerably more efficient than the composite likelihood estimator in a frequentist framework. From a Bayesian perspective, our approach opens the way for new techniques such as Bayesian model comparison in multivariate and spatial extremes.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 4813-4844.

Received: August 2017
First available in Project Euclid: 27 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 60G70: Extreme value theory; extremal processes 62F12: Asymptotic properties of estimators

Asymptotic normality Bayesian statistics efficient inference full likelihood Markov chain Monte Carlo max-stability multivariate extremes

Creative Commons Attribution 4.0 International License.


Dombry, Clément; Engelke, Sebastian; Oesting, Marco. Bayesian inference for multivariate extreme value distributions. Electron. J. Statist. 11 (2017), no. 2, 4813--4844. doi:10.1214/17-EJS1367. https://projecteuclid.org/euclid.ejs/1511773485

Export citation


  • P. Asadi, A. Davison, and S. Engelke. Extremes on river networks., Ann. Appl. Stat., 9 :2023–2050, 2015.
  • C. Berg, J. P. R. Christensen, and P. Ressel., Harmonic Analysis on Semigroups. Springer-Verlag, New York, 1984.
  • M.-O. Boldi and A. C. Davison. A mixture model for multivariate extremes., J. R. Stat. Soc. Ser. B Stat. Methodol., 69:217–229, 2007.
  • B. M. Brown and S. I. Resnick. Extreme values of independent stochastic processes., J. Appl. Probab., 14:732–739, 1977.
  • T. A. Buishand, L. de Haan, and C. Zhou. On spatial extremes: with application to a rainfall problem., Ann. Appl. Stat., 2:624–642, 2008.
  • S. Castruccio, R. Huser, and M. G. Genton. High-order composite likelihood inference for max-stable distributions and processes., J. Comput. Graph. Statist., 25 :1212–1229, 2016.
  • R. E. Chandler and S. Bate. Inference for clustered data using the independence loglikelihood., Biometrika, 94:167–183, 2007.
  • S. G. Coles and J. A. Tawn. Modelling extreme multivariate events., J. R. Stat. Soc. Ser. B Stat. Methodol., 53:377–392, 1991.
  • A. C. Davison and M. M. Gholamrezaee. Geostatistics of extremes., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468:581–608, 2012.
  • R. de Fondeville and A. Davison. High-dimensional peaks-over-threshold inference. Available from, https://arxiv.org/abs/1605.08558, 2017.
  • M. Denuit, J. Dhaene, M. Goovaerts, and R. Kaas., Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley, 2005.
  • C. Dombry and F. Éyi-Minko. Regular conditional distributions of continuous max-infinitely divisible random fields., Electron. J. Probab., 18:1–21, 2013.
  • C. Dombry, F. Eyi-Minko, and M. Ribatet. Conditional simulation of max-stable processes., Biometrika, 100:111–124, 2013.
  • C. Dombry, S. Engelke, and M. Oesting. Exact simulation of max-stable processes., Biometrika, 103:303–317, 2016.
  • C. Dombry, S. Engelke, and M. Oesting. Asymptotic properties of the maximum likelihood estimator for multivariate extreme value distributions. Available from, https://arxiv.org/abs/1612.05178v2, 2017a.
  • C. Dombry, M. Genton, R. Huser, and M. Ribatet. Full likelihood inference for max-stable data. Available from, https://arxiv.org/abs/1703.08665, 2017b.
  • J. H. J. Einmahl, A. Kiriliouk, A. Krajina, and J. Segers. An M–estimator of spatial tail dependence., J. R. Stat. Soc. Ser. B Stat. Methodol., 78:275–298, 2016.
  • S. Engelke and J. Ivanovs. Robust bounds in multivariate extremes., The Annals of Applied Probability, 2017. To appear.
  • S. Engelke, A. Malinowski, M. Oesting, and M. Schlather. Statistical inference for max-stable processes by conditioning on extreme events., Adv. Appl. Probab., 46:478–495, 2014.
  • S. Engelke, A. Malinowski, Z. Kabluchko, and M. Schlather. Estimation of Hüsler–Reiss distributions and Brown–Resnick processes., J. R. Stat. Soc. Ser. B Stat. Methodol., 77:239–265, 2015.
  • M. G. Genton, Y. Ma, and H. Sang. On the likelihood function of gaussian max-stable processes., Biometrika, 98:481–488, 2011.
  • M. Hofert and M. Mächler. Nested Archimedean copulas meet R: The nacopula package., J. Stat. Softw., 39:1–20, 2011.
  • R. Huser and A. Davison. Composite likelihood estimation for the brown-resnick process., Biometrika, 100:511–518, 2013.
  • R. Huser, A. C. Davison, and M. G. Genton. Likelihood estimators for multivariate extremes., Extremes, 19:79–103, 2015.
  • J. Hüsler and R.-D. Reiss. Maxima of normal random vectors: between independence and complete dependence., Statist. Probab. Lett., 7:283–286, 1989.
  • Z. Kabluchko, M. Schlather, and L. de Haan. Stationary max-stable fields associated to negative definite functions., Ann. Probab., 37 :2042–2065, 2009.
  • R. E. Kass and A. E. Raftery. Bayes factors., J. Am. Stat. Assoc., 90:773–795, 1995.
  • A. K. Nikoloulopoulos, H. Joe, and H. Li. Extreme value properties of multivariate t copulas., Extremes, 12:129–148, 2009.
  • P. J. Northrop and N. Attalides. Posterior propriety in Bayesian extreme value analyses using reference priors., Statist. Sinica, 26:721–743, 2016.
  • T. Opitz. Extremal $t$ processes: Elliptical domain of attraction and a spectral representation., J. Multivariate Anal., 122:409–413, 2013.
  • S. A. Padoan, M. Ribatet, and S. A. Sisson. Likelihood-based inference for max-stable processes., J. Am. Stat. Assoc., 105:263–277, 2010.
  • H. M. Ramos, J. Ollero, and S. M. A. A sufficient condition for generalized lorenz order., Journal of Economic Theory, 90:286–292, 2000.
  • M. Ribatet, D. Cooley, and A. C. Davison. Bayesian inference from composite likelihoods, with an application to spatial extremes., Statistica Sinica, 22:813–845, 2012.
  • H. Rootzén and N. Tajvidi. Multivariate generalized Pareto distributions., Bernoulli, 12:917–930, 2006.
  • H. Rootzén, J. Segers, and J. L. Wadsworth. Multivariate peaks over thresholds models., Extremes, 2017.
  • M. Schlather. Models for stationary max-stable random fields., Extremes, 5:33–44, 2002.
  • D. Shi. Fisher information for a multivariate extreme value distribution., Biometrika, 82:644–649, 1995.
  • A. Stephenson and J. Tawn. Bayesian inference for extremes: Accounting for the three extremal types., Extremes, 7:291–307, 2005a.
  • A. Stephenson and J. A. Tawn. Exploiting occurrence times in likelihood inference for componentwise maxima., Biometrika, 92:213–227, 2005b.
  • E. Thibaud and T. Opitz. Efficient inference and simulation for elliptical Pareto processes., Biometrika, 102:855–870, 2015.
  • E. Thibaud, J. Aalto, D. S. Cooley, A. C. Davison, and J. Heikkinen. Bayesian inference for the Brown–Resnick process, with an application to extreme low temperatures., Ann. Appl. Stat., 10 :2303–2324, 2016.
  • A. W. van der Vaart., Asymptotic statistics. Cambridge University Press, Cambridge, 1998.
  • J. L. Wadsworth. On the occurrence times of componentwise maxima and bias in likelihood inference for multivariate max-stable distributions., Biometrika, 102:705–711, 2015.
  • J. L. Wadsworth and J. A. Tawn. Efficient inference for spatial extreme value processes associated to log-gaussian random functions., Biometrika, 101:1–15, 2014.