Since many developments to the functional extreme value theory have been made during the last decades, this paper reviews recent results on max-stable processes and covers a large range of themes such as finite dimensional distributions, parametric models, dependence measure, inferential procedure, model selection and (conditional) simulations. An application to the spatial modeling of wind gusts in Netherlands is given.
De nombreux progrès ont été accomplis ces dernières décennies sur la théorie des valeurs extrêmes fonctionnelle. Dans ce papier nous regroupons les résultats principaux concernant les processus max-stables. Ainsi cette revue de littérature couvre une gamme variée de domaines : lois fini-dimensionnelles, modèles paramétriques, mesures de dépendance, procédure inférentielles, sélection de modèles et simulations (conditionnelles). Une application à la modélisation spatiale des rafales de vents aux Pays-bas est donnée.
Mot clés : Processus max-stable, Fonction du coefficient extrême, Vraisemblance composite, Simulation
@article{JSFS_2013__154_2_156_0, author = {Ribatet, Mathieu}, title = {Spatial extremes: {Max-stable} processes at work}, journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique}, pages = {156--177}, publisher = {Soci\'et\'e fran\c{c}aise de statistique}, volume = {154}, number = {2}, year = {2013}, mrnumber = {3120441}, zbl = {1316.62141}, language = {en}, url = {http://www.numdam.org/item/JSFS_2013__154_2_156_0/} }
TY - JOUR AU - Ribatet, Mathieu TI - Spatial extremes: Max-stable processes at work JO - Journal de la société française de statistique PY - 2013 SP - 156 EP - 177 VL - 154 IS - 2 PB - Société française de statistique UR - http://www.numdam.org/item/JSFS_2013__154_2_156_0/ LA - en ID - JSFS_2013__154_2_156_0 ER -
Ribatet, Mathieu. Spatial extremes: Max-stable processes at work. Journal de la société française de statistique, Volume 154 (2013) no. 2, pp. 156-177. http://www.numdam.org/item/JSFS_2013__154_2_156_0/
[1] S. Aulbach, M. Falk, and M. Hofmann. On max-stable processes and the functional D-norm. Extremes, pages 1–29, 2012. | MR | Zbl
[2] B. M. Brown and S. I. Resnick. Extreme values of independent stochastic processes. Journal of Applied Probability, 14:732–739, 1977. | MR | Zbl
[3] T. A. Buishand, L. de Haan, and C. Zhou. On spatial extremes: With application to a rainfall problem. Annals Of Applied Statistics, 2(2):624–642, June 2008. | MR | Zbl
[4] R. E. Chandler and S. Bate. Inference for clustered data using the independence loglikelihood. Biometrika, 94(1):167–183, 2007. | MR | Zbl
[5] D. Cooley, P. Naveau, and P. Poncet. Variograms for spatial max-stable random fields. In P. Bertail, P. Soulier, P. Doukhan, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, and S. Zeger, editors, Dependence in Probability and Statistics, volume 187 of Lecture Notes in Statistics, pages 373–390. Springer New York, 2006. | MR | Zbl
[6] A.C. Davison, S.A. Padoan, and M. Ribatet. Statistical modelling of spatial extremes. Statistical Science, 7(2):161–186, 2012. | MR
[7] L. de Haan. A spectral representation for max-stable processes. The Annals of Probability, 12(4):1194–1204, 1984. | MR | Zbl
[8] L. de Haan and A. Ferreira. The Generalized Pareto process; with application. Submitted, 2012. Preprint http://arxiv.org/abs/1203.2551, 2012. | MR
[9] Laurens de Haan and Ana Fereira. Extreme value theory: An introduction. Springer Series in Operations Research and Financial Engineering, 2006. | MR | Zbl
[10] C. Dombry and F. Éyi-Minko. Regular conditional distributions of max infinitely divisible random fields. Electronic Journal of Probability, 18(7):1–21, 2013. | MR | Zbl
[11] C. Dombry, F. Éyi-Minko, and M. Ribatet. Conditional simulations of max-stable processes. Biometrika, 100(1):111–124, 2013. | MR
[12] C. Dombry and M. Ribatet. Functional regular variations, Pareto processes and peaks over threshold. Submitted, 2013. Preprint www.math.univ-montp2.fr/~ribatet/docs/Dombry2012.pdf. | MR
[13] R. J. Erhardt and R. L. Smith. Approximate Bayesian computing for spatial extremes. Computational Statistics and Data Analysis, 56(6):1468–1481, 2012. | MR | Zbl
[14] M. G. Genton, Y. Ma, and H. Sang. On the likelihood function of Gaussian max-stable processes. Biometrika, 98(2):481–488, 2011. | MR | Zbl
[15] R. Huser and A. C. Davison. Composite likelihood estimation for the Brown–Resnick process. Biometrika, 100(2):511–518, 2013. | MR
[16] Z. Kabluchko, M. Schlather, and L. de Haan. Stationary max-stable fields associated to negative definite functions. Annals of Probability, 37(5):2042–2065, 2009. | MR | Zbl
[17] J. T. Kent. Robust properties of likelihood ratio tests. Biometrika, 69:19–27, 1982. | MR | Zbl
[18] A.K. Nikoloulopoulos, H. Joe, and H. Li. Extreme values properties of multivariate copulas. Extremes, 12:129–148, 2009. | MR | Zbl
[19] M. Oesting, Z. Kabluchko, and M. Schlather. Simulation of Brown–Resnick processes. Extremes, 15:89–107, 2012. 10.1007/s10687-011-0128-8. | MR
[20] T. Opitz. Extremal- process: Elliptical domain of attraction and a spectral representation. Submitted, 2012. Preprint http://arxiv.org/abs/1207.2296. | MR | Zbl
[21] S.A. Padoan, M. Ribatet, and S. Sisson. Likelihood-based inference for max-stable processes. Journal of the American Statistical Association (Theory & Methods), 105(489):263–277, 2010. | MR
[22] M. D. Penrose. Semi-min-stable processes. Annals of Probability, 20(3):1450–1463, 1992. | MR | Zbl
[23] M. Ribatet, D. Cooley, and A.C. Davison. Bayesian inference from composite likelihoods, with an application to spatial extremes. Statistica Sinica, 22:813–845, 2012. | MR | Zbl
[24] M. Ribatet and M. Sedki. Extreme value copulas and max-stable processes. To appear in Journal de la Société Française de Statistique, 2013. | MR
[25] M. Ribatet, R. Singleton, and R Core Team. SpatialExtremes: Modelling Spatial Extremes, 2013. R package version 2.0-1.
[26] A. Rotnitzky and N. Jewell. Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data. Biometrika, 77:495–497, 1990. | MR | Zbl
[27] M. Schlather. Models for stationary max-stable random fields. Extremes, 5(1):33–44, March 2002. | MR | Zbl
[28] M. Schlather, P. Menck, R. Singleton, and R Core Team. RandomFields: Simulation and Analysis of Random Fields, 2013. R package version 2.0.66.
[29] M. Schlather and J.A. Tawn. A dependence measure for multivariate and spatial extremes: Properties and inference. Biometrika, 90(1):139–156, 2003. | MR | Zbl
[30] R. L. Smith. Max-stable processes and spatial extreme. Unpublished manuscript, 1990.
[31] C. Varin, N. Reid, and D. Firth. An overview of composite likelihood methods. Statistica Sinica, 21(5–42), 2011. | MR
[32] C. Varin and P. Vidoni. A note on composite likelihood inference and model selection. Biometrika, 92(3):519–528, 2005. | MR | Zbl
[33] Y. Wang and S. A. Stoev. Conditional sampling for spectrally discrete max-stable random fields. Advances in Applied Probability, 443:461–483, 2011. | MR | Zbl