Advertisement

Extremes

, Volume 20, Issue 4, pp 807–838 | Cite as

Clustering of high values in random fields

  • Luísa Pereira
  • Ana Paula Martins
  • Helena Ferreira
Article

Abstract

The asymptotic results that underlie applications of extreme random fields often assume that the variables are located on a regular discrete grid, identified with \(\mathbb {Z}^{2}\), and that they satisfy stationarity and isotropy conditions. Here we extend the existing theory, concerning the asymptotic behavior of the maximum and the extremal index, to non-stationary and anisotropic random fields, defined over discrete subsets of \(\mathbb {R}^{2}\). We show that, under a suitable coordinatewise mixing condition, the maximum may be regarded as the maximum of an approximately independent sequence of submaxima, although there may be high local dependence leading to clustering of high values. Under restrictions on the local path behavior of high values, criteria are given for the existence and value of the spatial extremal index which plays a key role in determining the cluster sizes and quantifying the strength of dependence between exceedances of high levels. The general theory is applied to the class of max-stable random fields, for which the extremal index is obtained as a function of well-known tail dependence measures found in the literature, leading to a simple estimation method for this parameter. The results are illustrated with non-stationary Gaussian and 1-dependent random fields. For the latter, a simulation and estimation study is performed.

Keywords

Random field Max-stable process Extremal dependence Spatial extremal index 

AMS 2000 Subject Classifications

60G60 60G70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

We are grateful to the referees for their detailed comments and important suggestions which significantly improved this paper. This research was supported by National Foundation of Science and Technology through UID/MAT/00212/2013.

References

  1. Adler, R.A.: The Geometry of Random Fields. Wiley, New York (1981)zbMATHGoogle Scholar
  2. Berman, S.: Sojourns and Extremes of Stochastic Processes. Taylor & Francis Ltd, USA (1992)Google Scholar
  3. Choi, H.: Almost sure limit theorem for stationary Gaussian random fields. J. Korean Stat. Soc. 39, 475–482 (2010)MathSciNetzbMATHGoogle Scholar
  4. Cooley, D., Naveau, P., Poncet, P.: Variograms for spatial max-stable random fields. Lect. Notes Stat. 187, 373–390 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204 (1984)Google Scholar
  6. Ferreira, M.: Estimating the extremal index through the tail dependence concept. Discussiones Mathematicae - Probab. Stat. 35, 61–74 (2015)MathSciNetCrossRefGoogle Scholar
  7. Ferreira, H., Pereira, L: How to compute the extremal index of stationary random fields. Stat. Probab. Lett. 78(11), 1301–1304 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ferreira, M., Ferreira, H.: On extremal dependence of block vectors. Kybernetika 48(5), 988–1006 (2012)MathSciNetzbMATHGoogle Scholar
  9. Ferreira, H., Pereira, L.: Point processes of exceedances by random fields. J. Stat. Plan. Infer. 142(3), 773–779 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Krajina, A.: An M-Estimator of Multivariate Dependence Concepts. PhD Thesis. Tilburg, Tilburg University Press (2010)Google Scholar
  11. Leadbetter, M.R., Lindgren, G., Rootzén, H: Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  12. Leadbetter, M.R., Nandagopalan, S.: On exceedance point process for stationary sequences under mild oscillation restrictions. In: Hüsler, J., Reiss, D (eds.) Extreme Value Theory: Proceedings, Oberwolfach, vol. 1987, pp 69–80. Springer, New York (1988)Google Scholar
  13. Leadbetter, M.R., Rootzén, H.: On Extreme Values in Stationary Random Fields. Stochastic Processes and Related Topics, 275-285 Trends Math. Birkhauser, Boston (1998)zbMATHGoogle Scholar
  14. Li, H.: Orthant tail dependence of multivariate etreme value distributions. J. Multivar. Anal. 100(1), 243–256 (2009)CrossRefzbMATHGoogle Scholar
  15. Pereira, L., Ferreira, H.: Limiting crossing probabilities of random fields. J. Appl. Probab. 3, 884–891 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Pereira, L.: On the extremal behavior of a non-stationary normal random field. J. Stat. Plan. Infer. 140(11), 3567–3576 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Piterbarg, V.I.: Asymptotics Methods in Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs, vol. 48. American Mathematical Society (1996)Google Scholar
  18. Schlather, M., Tawn, J.: A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90, 139–156 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Sibuya, M.: Bivariate extreme statistics. Ann. Inst. Stat. Math. 11(2), 195–210 (1959)MathSciNetzbMATHGoogle Scholar
  20. Smith, R.L.: Max-Stable Processes and Spatial Extremes. Preprint Univ North Carolina, USA (1990)Google Scholar
  21. Oliveira, J.T.: Structure theory of bivariate extremes: Extensions. Estudos de Matemática, Estatística e Econometria 7, 165–195 (1962/63)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Luísa Pereira
    • 1
  • Ana Paula Martins
    • 1
  • Helena Ferreira
    • 1
  1. 1.Universidade da Beira InteriorCentro de Matemática e Aplicações (CMA-UBI)CovilhãPortugal

Personalised recommendations