Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T13:11:30.281Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized max-linear models and their statistical interpolation

Part of: Stochastic processes

Published online by Cambridge University Press:  30 March 2016

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

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose a method to generate a max-stable process in C[0, 1] from a max-stable random vector in Rd by generalizing the max-linear model established by Wang and Stoev (2011). For this purpose, an interpolation technique that preserves max-stability is proposed. It turns out that if the random vector follows some finite-dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean-squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, a reasonable prediction of max-stable processes in space becomes possible. A possible extension to arbitrary dimensions is outlined.

Keywords

Multivariate extreme value distributionmultivariate generalized Pareto distributionmax-stable processgeneralized Pareto processD-normmax-linear modelprediction of max-stable processprediction of generalized Pareto process

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 736 - 751
Copyright
Copyright © 2015 by the Applied Probability Trust 

References

Aulbach, S., Falk, M. and Hofmann, M. (2013). On max-stable processes and the functional D-norm. Extremes 16 255-283.Google Scholar
Buishand, T. A., De Haan, L. and Zhou, C. (2008). On spatial extremes: with application to a rainfall problem. Ann. Appl. Statist. 2 624-642.Google Scholar
Davis, R. A. and Mikosch, T. (2008). Extremal value theory for space-time processes with heavy-tailed distributions. Stoch. Process. Appl. 118, 560–584.Google Scholar
De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12 1194-1204.Google Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. Available at http://people.few.eur.nl/ldehaan/EVTbook.correction.pdf and http://home.isa.utl.pt/∼anafh/corrections.pdf for corrections and extensions.Google Scholar
De Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in C[0,1]. Ann. Prob. 29 467-483.Google Scholar
De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40 317-337.Google Scholar
Dombry, C., Éyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100, 111–124.Google Scholar
Falk, M., Hofmann, M. and Zott, M. (2014). On generalized max-linear models and their statistical interpolation. Preprint. Available at http://arxiv.org/abs/1303.2602v2.Google Scholar
Falk, M., Hüsler, J. and Reiss, R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events, 3rd edn. Springer, Basel.Google Scholar
Ferreira, A. and De Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20 1717-1737.Google Scholar
Giné, E., Hahn, M. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115 249-274.Google Scholar
Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80 121-140.Google Scholar
Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes 12 401-424.CrossRefGoogle Scholar
Pickands, J. III (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49 859-878, 894–902.Google Scholar
Smith, R. L. (1990). Max-stable processes and spatial extremes. Preprint, University of Surrey. Available at http://www.stat.unc.edu/faculty/rs/papers/RLS_Papers.html.Google Scholar
Stoev, S. A. and Taqqu, M. S. (2005). Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes. Extremes 8 237-266.Google Scholar
Wang, Y. and Stoev, S. A. (2010). On the structure and representations of max-stable processes. Adv. Appl. Prob. 42 855-877.Google Scholar
Wang, Y. and Stoev, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Adv. Appl. Prob. 43 461-483.Google Scholar