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Statistical Inference for Max-Stable Processes by Conditioning on Extreme Events

  • Sebastian Engelke (a1), Alexander Malinowski (a2), Marco Oesting (a3) and Martin Schlather (a3)
Abstract

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

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Corresponding author
Postal address: Université de Lausanne, UNIL-Dorigny, Bâtiment Extranef, 1015 Lausanne, Switzerland. Email address: sebastian.engelke@unil.ch
∗∗ Postal address: Institut für Mathematik, Universität Mannheim, A5, 6, 68131 Mannheim, Germany.
∗∗ Email address: malinows@math.uni-goettingen.de
∗∗∗∗ Postal address: INRA, UMR 518 Math. Info. Appli., Rue Claude Bernard, 75005 Paris, France. Email address: marco.oesting@agroparistech.fr
∗∗∗∗∗ Email address: schlather@math.uni-mannheim.de
References
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[1] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stoch. Process. Appl. 119, 10551080. (Erratum: 121 (2011), 896–898.)
[2] Blanchet, J. and Davison, A. C. (2011). Spatial modeling of extreme snow depth. Ann. Appl. Statist. 5, 16991725.
[3] Brown, B. M. and Resnick, S. I. (1977). Extreme values of independent stochastic processes. J. Appl. Prob. 14, 732739.
[4] Cai, J.-J., Einmahl, J. H. J. and de Haan, L. (2011). Estimation of extreme risk regions under multivariate regular variation. Ann. Statist. 39, 18031826.
[5] Cooley, D., Davis, R. A. and Naveau, P. (2012). Approximating the conditional density given large observed values via a multivariate extremes framework, with application to environmental data. Ann. Appl. Statist. 6, 14061429.
[6] Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes. Proc. R. Soc. London 468, 581608.
[7] De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 11941204.
[8] De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.
[9] De Haan, L. and Pereira, T. T. (2006). Spatial extremes: models for the stationary case. Ann. Statist. 34, 146168.
[10] Dombry, C. and Ribatet, M. (2014). Functional regular variation, pareto processes and peaks over threshold. To appear in Statist. Interface.
[11] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Springer, Berlin.
[12] Engelke, S., Kabluchko, Z. and Schlather, M. (2011). An equivalent representation of the Brown–Resnick process. Statist. Prob. Lett. 81, 11501154.
[13] Engelke, S., Malinowski, A., Kabluchko, Z. and Schlather, M. (2014). Estimation of Hüsler–Reiss distributions and Brown–Resnick processes. To appear in J. R. Statist. Soc. B. Available at http://uk.arxiv.org/abs/1207.6886.
[14] Falk, M. and Tichy, D. (2012). Asymptotic conditional distribution of exceedance counts. Adv. Appl. Prob. 44, 270291.
[15] Ferreira, A. and de Haan, L. (2012). The generalized Pareto process; with a view towards application and simulation. Preprint. Available at http://uk.arxiv.org/abs/1203.2551.
[16] Gumbel, E. J. (1960). Distributions des valeurs extrêmes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9, 171173.
[17] Heffernan, J. E. and Tawn, J. A. (2004). A conditional approach for multivariate extreme values. With discussions and reply by the authors. J. R. Statist. Soc. Ser. B 66, 497546.
[18] Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: between independence and complete dependence. Statist. Prob. Lett. 7, 283286.
[19] Kabluchko, Z. (2011). Extremes of independent Gaussian processes. Extremes 14, 285310.
[20] Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Prob. 37, 20422065.
[21] Leadbetter, M. R. (1991). On a basis for ‘peaks over threshold’ modeling. Statist. Prob. Lett. 12, 357362.
[22] Meinguet, T. (2012). Maxima of moving maxima of continuous functions. Extremes 15, 267297.
[23] Meinguet, T. and Segers, J. (2010). Regularly varying time series in Banach spaces. Preprint. Available at http://uk.arxiv.org/abs/1001.3262.
[24] Oesting, M., Kabluchko, Z. and Schlather, M. (2012). Simulation of Brown–Resnick processes. Extremes 15, 89107.
[25] Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105, 263277.
[26] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York.
[27] Resnick, S. I. and Roy, R. (1991). Random usc functions, max-stable processes and continuous choice. Ann. Appl. Prob. 1, 267292.
[28] Rootzén, H. and Tajvidi, N. (2006). Multivariate generalized Pareto distributions. Bernoulli 12, 917930.
[29] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5, 3344.
[30] Schlather, M. and Tawn, J. A. (2003). A dependence measure for multivariate and spatial extreme values: Properties and inference. Biometrika 90, 139156.
[31] Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript. Available at http://www.unc.edu/~rls/.
[32] Wang, Y. and Stoev, S. A. (2010). On the structure and representations of max-stable processes. Adv. Appl. Prob. 42, 855877.
[33] Whitt, W. (1970). Weak convergence of probability measures on the function space C[0,∞). Ann. Math. Statist. 41, 939944.
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Advances in Applied Probability
  • ISSN: 0001-8678
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