Statistics and Computing

, Volume 24, Issue 4, pp 651–662 | Cite as

Estimation of spatial max-stable models using threshold exceedances



Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block maxima, typically annual, is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. We focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by Ledford and Tawn (Biometrika 83:169–187, 1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzén and Tajvidi in Bernoulli 12:917–930, 2006) of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study where both processes in a domain of attraction of a max-stable process and max-stable processes are successively considered as time replications, according to different degrees of spatial dependency. Results put forward how the nature of the time replications influences the bias of estimations and highlight the choice of each approach regarding to the strength of the spatial dependencies and the threshold choice.


Composite likelihood Extremal dependence Generalized Pareto distribution Spatial statistics 


  1. Bacro, J.N., Gaetan, C.: A review on spatial extreme modelling. In: Porcu, E., Montero, J.M., Schlather, M. (eds.) Advances and Challenges in Space-Time Modelling of Natural Events, pp. 103–124. Springer, New York (2012) CrossRefGoogle Scholar
  2. Beirlant, B.J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004) CrossRefGoogle Scholar
  3. Bel, L., Bacro, J.N., Lantuéjoul, C.: Assessing extremal dependence of environmental spatial fields. Environmetrics 19, 163–182 (2008) CrossRefMathSciNetGoogle Scholar
  4. Bevilacqua, M., Gaetan, C., Mateu, J., Porcu, E.: Estimating space and space-time covariance functions: a weighted composite likelihood approach. J. Am. Stat. Assoc. 107, 268–280 (2012) CrossRefMATHMathSciNetGoogle Scholar
  5. Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008) CrossRefMATHMathSciNetGoogle Scholar
  6. Carlstein, A.: The use of subseries values for estimating the variance of a general statistic from a stationary sequence. Ann. Stat. 14, 1171–1179 (1986) CrossRefMATHMathSciNetGoogle Scholar
  7. Casson, E., Coles, S.G.: Spatial regression models for extremes. Extremes 1, 449–468 (1999) CrossRefMATHGoogle Scholar
  8. Coles, S.G.: Regional modelling of extreme storms via max-stable processes. J. R. Stat. Soc., Ser. B 55, 797–816 (1993) MATHMathSciNetGoogle Scholar
  9. Coles, S.G.: An Introduction to Statistical Modeling of Extreme Value. Springer, Boca Raton (2001) CrossRefGoogle Scholar
  10. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. J. R. Stat. Soc., Ser. B 53, 377–392 (1991) MATHMathSciNetGoogle Scholar
  11. Coles, S.G., Tawn, J.A.: Statistical methods models for multivariate extremes: an application to structural design. Appl. Stat. 43, 1–48 (1994) CrossRefMATHGoogle Scholar
  12. Coles, S.G., Tawn, J.A.: Modelling extremes of the areal rainfall process. J. R. Stat. Soc., Ser. B 58, 329–347 (1996) MATHMathSciNetGoogle Scholar
  13. Davis, R., Yau, C.-Y.: Comments on pairwise likelihood in time series models. Stat. Sin. 21, 255–277 (2011) MATHMathSciNetGoogle Scholar
  14. Davison, A.C., Gholamrezaee, M.M.: Geostatistics of extremes. Proc. R. Soc. Lond., Ser. A 468, 581–608 (2012) CrossRefMathSciNetGoogle Scholar
  15. Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc., Ser. B 3, 393–442 (1990) MathSciNetGoogle Scholar
  16. Davison, A.C., Padoan, S.A., Ribatet, M.: Statistical modelling of spatial extremes. Stat. Sci. 27, 161–186 (2012) CrossRefMathSciNetGoogle Scholar
  17. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204 (1984) CrossRefMATHMathSciNetGoogle Scholar
  18. Demarta, S., McNeil, A.J.: The t copula and related copulas. Int. Stat. Rev. 73, 111–129 (2005) CrossRefMATHGoogle Scholar
  19. Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18, 173–188 (2007) CrossRefMathSciNetGoogle Scholar
  20. Genton, M.G., Ma, Y., Sang, H.: On the likelihood function of Gaussian max-stable processes. Biometrika 98, 481–488 (2011) CrossRefMATHMathSciNetGoogle Scholar
  21. Guyon, X.: Random Fields on a Network. Springer, New York (1995) MATHGoogle Scholar
  22. Heyde, C.: Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer, New York (1997) CrossRefMATHGoogle Scholar
  23. Huser, R., Davison, A.C.: Space-time modelling of extreme events. Technical report (2012). arXiv:1201.3245
  24. Jeon, S., Smith, R.: Dependence structure of spatial extremes using threshold approach. Technical report (2012). arXiv:1209.6344
  25. Kabluchko, Z., Schlather, M., de Haan, L.: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37, 2042–2065 (2009) CrossRefMATHMathSciNetGoogle Scholar
  26. Kuk, A.: A hybrid pairwise likelihood method. Biometrika 94, 939–952 (2007) CrossRefMATHMathSciNetGoogle Scholar
  27. Lantuéjoul, C., Bacro, J.-N., Bel, L.: Storm processes and stochastic geometry. Extremes 14, 413–428 (2011) CrossRefMathSciNetGoogle Scholar
  28. Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996) CrossRefMATHMathSciNetGoogle Scholar
  29. Lindsay, B.: Composite likelihood methods. Contemp. Math. 80, 221–239 (1988) CrossRefMathSciNetGoogle Scholar
  30. Madsen, H., Rasmussen, P.F., Rosbjerg, D.: Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events: 1. at-site modeling. Water Resour. Res. 33, 747–757 (1997) CrossRefGoogle Scholar
  31. Nadarajah, S.: Multivariate declustering techniques. Environmetrics 12, 357–365 (2001) CrossRefGoogle Scholar
  32. Nikoloulopoulos, A., Joe, H., Li, H.: Extreme value properties of multivariate t-copulas. Extreme 12, 129–148 (2009) CrossRefMATHMathSciNetGoogle Scholar
  33. Opitz, T.: A spectral construction of the extremal t process. Technical report (2012). arXiv:1207.2296
  34. Padoan, S., Bevilacqua, M.: CompRandFld: Composite-Likelihood Based Analysis of Random Fields. R package version 1.0.2 (2012) Google Scholar
  35. Padoan, S.A., Ribatet, M., Sisson, S.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105, 263–277 (2010) CrossRefMathSciNetGoogle Scholar
  36. Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975) CrossRefMATHMathSciNetGoogle Scholar
  37. Rakonczai, P., Tajvidi, N.: On prediction on bivariate extremes. Int. J. Intell. Technol. Appl. Stat. 3, 115–139 (2010) Google Scholar
  38. Resnick, S.: Extreme Values, Regular Variation and Point Processes. Springer, New York (1987) CrossRefMATHGoogle Scholar
  39. Ribatet, M.: SpatialExtremes: modelling spatial extremes. R package version 1.9-0 (2012) Google Scholar
  40. Rootzén, H., Tajvidi, N.: Multivariate generalized Pareto distributions. Bernoulli 12, 917–930 (2006) CrossRefMATHMathSciNetGoogle Scholar
  41. Schlather, M.: Models for stationary max-stable random fields. Extremes 5, 33–44 (2002) CrossRefMATHMathSciNetGoogle Scholar
  42. Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90, 139–156 (2003) CrossRefMATHMathSciNetGoogle Scholar
  43. Shang, H., Yan, J., Zhang, X.: A two-step composite likelihood approach with data fusion for max-stable processes in spatial extremes modelling. Technical report (2012). arXiv:1204.0286
  44. Smith, R.L.: Max-stable processes and spatial extremes. Preprint, University of Surrey (1990a) Google Scholar
  45. Smith, R.L.: Regional estimation from spatially dependent data. Preprint, University of North Carolina (1990b) Google Scholar
  46. Smith, E.L., Stephenson, A.G.: An extended Gaussian max-stable process model for spatial extremes. J. Stat. Plan. Inference 139, 1266–1275 (2009) CrossRefMATHMathSciNetGoogle Scholar
  47. Tawn, J.: Bivariate extreme value theory: models and estimation. Biometrika 75, 397–415 (1988) CrossRefMATHMathSciNetGoogle Scholar
  48. Tawn, J.: Modelling multivariate extreme value distributions. Biometrika 77, 245–253 (1990) CrossRefMATHGoogle Scholar
  49. Turkman, K., Amaral Turkman, M., Pereira, J.: Asymptotic models and inference for extremes of spatio-temporal data. Extremes 13, 375–397 (2010) CrossRefMATHMathSciNetGoogle Scholar
  50. Varin, C., Reid, N., Firth, D.: An overview of composite likelihood methods. Stat. Sin. 21, 5–42 (2011) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.I3MUniversité Montpellier IIMontpellierFrance
  2. 2.DAISUniversità Ca’ Foscari - VeneziaVeneziaItaly

Personalised recommendations