Journal of Applied Probability

On generalized max-linear models and their statistical interpolation

Michael Falk, Martin Hofmann, and Maximilian Zott

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Abstract

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.

Article information

Source
J. Appl. Probab. Volume 52, Number 3 (2015), 736-751.

Dates
First available in Project Euclid: 22 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1445543843

Digital Object Identifier
doi:10.1239/jap/1445543843

Mathematical Reviews number (MathSciNet)
MR3414988

Zentralblatt MATH identifier
1336.60101

Subjects
Primary: 60G70: Extreme value theory; extremal processes

Keywords
Multivariate extreme value distribution multivariate generalized Pareto distribution max-stable process generalized Pareto process D-norm max-linear model prediction of max-stable process prediction of generalized Pareto process

Citation

Falk, Michael; Hofmann, Martin; Zott, Maximilian. On generalized max-linear models and their statistical interpolation. J. Appl. Probab. 52 (2015), no. 3, 736--751. doi:10.1239/jap/1445543843. https://projecteuclid.org/euclid.jap/1445543843.


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References

  • Aulbach, S., Falk, M. and Hofmann, M. (2013). On max-stable processes and the functional $D$-norm. Extremes 16, 255–283.
  • 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.
  • Davis, R. A. and Mikosch, T. (2008). Extremal value theory for space-time processes with heavy-tailed distributions. Stoch. Process. Appl. 118, 560–584.
  • De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204.
  • 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/$\sim$anafh/corrections.pdf for corrections and extensions.
  • De Haan, L. and Lin, T. (2001). On convergence toward an extreme value distribution in $C[0,1]$. Ann. Prob. 29, 467–483.
  • De Haan, L. and Resnick, S. I. (1977). Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitsth. 40, 317–337.
  • Dombry, C., Éyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100, 111–124.
  • 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.
  • Falk, M., Hüsler, J. and Reiss, R.-D. (2011). Laws of Small Numbers: Extremes and Rare Events, 3rd edn. Springer, Basel.
  • Ferreira, A. and De Haan, L. (2014). The generalized Pareto process; with a view towards application and simulation. Bernoulli 20, 1717–1737.
  • Giné, E., Hahn, M. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165.
  • Hult, H. and Lindskog, F. (2005). Extremal behavior of regularly varying stochastic processes. Stoch. Process. Appl. 115, 249–274.
  • Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N. S.) 80, 121–140.
  • Kabluchko, Z. (2009). Spectral representations of sum- and max-stable processes. Extremes 12, 401–424.
  • Pickands, J., III (1981). Multivariate extreme value distributions. Bull. Inst. Internat. Statist. 49, 859–878, 894–902.
  • 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.
  • Stoev, S. A. and Taqqu, M. S. (2005). Extremal stochastic integrals: a parallel between max-stable processes and $\alpha$-stable processes. Extremes 8, 237–266.
  • Wang, Y. and Stoev, S. A. (2010). On the structure and representations of max-stable processes. Adv. Appl. Prob. 42, 855–877.
  • Wang, Y. and Stoev, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Adv. Appl. Prob. 43, 461–483. \endharvreferences