Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T10:45:31.240Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized max-linear models in max-stable random fields

Part of: Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  15 September 2017

Michael Falk  and
Maximilian Zott
Michael Falk*
Affiliation:
University of Würzburg
Maximilian Zott*
Affiliation:
University of Würzburg
*
* Postal address: Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany.
* Postal address: Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

In practice, it is not possible to observe a whole max-stable random field. Therefore, we propose a method to reconstruct a max-stable random field in C([0, 1]k) by interpolating its realizations at finitely many points. The resulting interpolating process is again a max-stable random field. This approach uses a generalized max-linear model. Promising results have been established in the k = 1 case of Falk et al. (2015). However, the extension to higher dimensions is not straightforward since we lose the natural order of the index space.

Keywords

Multivariate extreme value distributionmax-stable random fieldD-normmax-linear modelstochastic interpolation

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62M40: Random fields; image analysis
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 797 - 810
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aulbach, S., Falk, M. and Hofmann, M. (2013). On max-stable processes and the functional D-norm. Extremes 16, 255–283. CrossRefGoogle Scholar
[2] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. John Wiley, Chichester. CrossRefGoogle Scholar
[3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. Google Scholar
[4] Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Prob. 14, 732–739. Google Scholar
[5] Cooley, D. and Sain, S. R. (2012). Discussion of 'Statistical modeling of spatial extremes' by A. C. Davison, S. A. Padoan and M. Ribatet. Statist. Sci. 27, 187–188. CrossRefGoogle Scholar
[6] Davison, A. C., Padoan, S. A. and Ribatet, M. (2012). Statistical modeling of spatial extremes. Statist. Sci. 27, 161–186. Google Scholar
[7] Davison, A. C., Padoan, S. A. and Ribatet, M. (2012). Rejoinder: 'Statistical modeling of spatial extremes'. Statist. Sci. 27, 199–201. Google Scholar
[8] De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. See http://people.few.eur.nl/ldehaan/EVTbook.correction.pdf and http://home.isa.utl.pt/~anafh/corrections.pdf for corrections and extensions. Google Scholar
[9] De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337. CrossRefGoogle Scholar
[10] Dombry, C., Éyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100, 111–124. Google Scholar
[11] Falk, M., Hofmann, M. and Zott, M. (2015). On generalized max-linear models and their statistical interpolation. J. Appl. Prob. 52, 736–751. CrossRefGoogle Scholar
[12] Falk, M., Hüsler, J. and Reiss, R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events, 3rd edn. Birkhäuser, Basel. CrossRefGoogle Scholar
[13] Gabda, D., Towe, R., Wadsworth, J. and Tawn, J. (2012). Discussion of 'Statistical modeling of spatial extremes' by A. C. Davison, S. A. Padoan and M. Ribatet. Statist. Sci. 27, 189–192. Google Scholar
[14] Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165. Google Scholar
[15] Huser, R. and Davison, A. C. (2013). Composite likelihood estimation for the Brown–Resnick process. Biometrika 100, 511–518. Google Scholar
[16] Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes 12, 401–424. CrossRefGoogle Scholar
[17] Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Prob. 37, 2042–2065. Google Scholar
[18] Pickands, J., III (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49, 859–878, 894–902. Google Scholar
[19] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York. Google Scholar
[20] Segers, J. (2012). Nonparametric inference for max-stable dependence. Statist. Sci. 27, 193–196. Google Scholar
[21] Shaby, B. and Reich, B. J. (2012). Discussion of 'Statistical modeling of spatial extremes' by A. C. Davison, S. A. Padoan and M. Ribatet. Statist. Sci. 27, 197–198. Google Scholar
[22] Wang, Y. and Stoev, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Adv. Appl. Prob. 43, 461–483. Google Scholar