Measuring and modelling multivariate and spatial dependence of extremes

Bacro, Jean-Noël; Toulemonde, Gwladys

Journal de la société française de statistique, Tome 154 (2013) no. 2, pp. 139-155.

Lorsque l’on s’intéresse aux extrêmes multivariés ou spatiaux, il est important d’identifier au mieux la nature des dépendances entre réalisations extrêmes. La notion d’indépendance asymptotique des extrêmes est à ce titre primordiale. Dans un cadre multivarié, différentes mesures quantifiant la dépendance extrêmale ont été proposées et des tests d’indépendance asymptotique mis en oeuvre. Pour le cadre spatial, la modélisation classique s’appuie sur les processus max-stables mais les structures de dépendance présentes peuvent être plus complexes que celles gérées par ce type de processus. Dans cet article, les principaux résultats sur l’inférence et la modélisation de la dépendance entre réalisations extrêmes sont présentés et l’accent est mis sur les modèles récents permettant une prise en compte de l’indépendance asymptotique.

Within both multivariate extreme models and spatial processes, it is important to precisely identify dependencies among extremes. In particular detecting asymptotic independence is fundamental and in a multivariate setting, many authors have proposed measures of extreme dependence/independence with sometimes associated tests. Dependence structures within spatial processes are more complex and very few authors have considered other structures than the max-stable one. In this survey paper some major contributions to inference and modelling for extremal dependencies are presented in multivariate and spatial contexts, including the presentation of recent spatial models allowing asymptotic independence.

[1] Ancona-Navarrete, M.A.; Tawn, J.A. Diagnostics for pairwise extremal dependence in spatial processes, Extremes, Volume 5 (2002), pp. 271-285 | MR | Zbl

[2] Bel, L.; Bacro, J. N.; Lantuéjoul, C. Assessing extremal dependence of environmental spatial fields, Environmetrics, Volume 19 (2008) no. 2, pp. 163-182 | MR

[3] Bacro, J. N.; Bel, L.; Lantuéjoul, C. Testing the independence of maxima: from bivariate vectors to spatial extreme fields, Extremes, Volume 13 (2010), pp. 155-175 | MR | Zbl

[4] Buishand, T. A.; de Haan, L.; Zhou, C. On spatial extremes: with application to a rainfall problem, The Annals of Applied Statistics, Volume 2 (2008), pp. 624-642 | MR | Zbl

[5] Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987 | MR | Zbl

[6] Beirlant, J.; Goegebeur, Y.; Teugels, J.; Segers, J. Statistics of extremes. Theory and application, Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester, 2004 | MR | Zbl

[7] Balkema, A. A.; Resnick, S. I. Max-infinite divisibility, Journal of Applied Probability, Volume 14 (1977), pp. 309-319 | MR | Zbl

[8] Buishand, T. A. Bivariate extreme value data and the station year method, Journal of Hydrology, Volume 69 (1984), pp. 77-95

[9] Capéraà, P.; Fougères, A.-L.; Genest., C. A nonparametric estimation procedure for bivariate extreme values copulas, Biometrika, Volume 84 (1997), pp. 567-577 | MR | Zbl

[10] Coles, S. G.; Heffernan, J. E.; Tawn, J. A. Dependence measures for extremes value analyses, Extremes, Volume 2 (1999), pp. 339-365 | Zbl

[11] Cooley, D.; Naveau, P.; Poncet, P. Variograms for spatial max-stable random fields, Dependence in probability and statistics (Lecture Notes in Statistic), Volume 187, Springer, New York, 2006, pp. 373-390 | MR | Zbl

[12] Coles, S. G. An introduction to statistical modeling of extreme values, Springer Series in Statistics, Springer-Verlag London Ltd., London, 2001 | MR | Zbl

[13] Coles, S. G. Regional modelling of extreme storms via max-stable processes, Journal of the Royal Statistical Society. Series B. Methodological, Volume 55 (1993), pp. 797-816 | MR | Zbl

[14] Coles, S. G.; Tawn, J. A. Modelling extremes of the areal rainfall process, Journal of the Royal Statistical Society. Series B. Methodological, Volume 58 (1996), pp. 329-347 | MR | Zbl

[15] de Haan, L. A spectral representation for max-stable processes, The Annals of Probability, Volume 12 (1984), pp. 1194-1204 | MR | Zbl

[16] de Carvalho, M.; Ramos, A. Bivariate extreme statistics, II, REVSTAT Statistical Journal, Volume 10 (2012), pp. 83-107 | MR | Zbl

[17] Draisma, G.; Drees, H.; Ferreira, A.; de Haan, L. Bivariate tail estimation: dependence in asymptotic independence, Bernoulli, Volume 10 (2004), pp. 251-280 | MR | Zbl

[18] Davison, A. C.; Gholamrezaee, M. M. Geostatistics of extremes, Proceedings of The Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences, Volume 468 (2012), pp. 581-608 | MR | Zbl

[19] de Haan, L.; de Ronde, J. Sea and Wind: Multivariate Extreme at Work, Extremes, Volume 40 (1998), pp. 7-45 | MR | Zbl

[20] de Haan, L.; Ferreira, A. Extreme value theory: an Introduction, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006 | MR | Zbl

[21] de Haan, L.; Resnick, S.I. Limit theory for multivariate sample extremes, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, Volume 1 (1977), pp. 317-337 | MR | Zbl

[22] Deheuvels, P.; Martynov, G. Cramer-Von Mises-type with applications to test of independence for multivariate extreme value distributions, Communications in Statistics - Theory and Methods, Volume 25 (1996), pp. 871-908 | MR | Zbl

[23] de Haan, L.; Pereira, T. T. Spatial extremes: models for the stationary case, The Annals of Statistics, Volume 34 (2006), pp. 146-168 | MR | Zbl

[24] Davison, A. C.; Padoan, S.; Ribatet, M. Statistical modelling of spatial extremes, Statistical Science, Volume 27 (2012), pp. 161-186 | MR | Zbl

[25] Frick, M.; Kaufmann, E.; Reiss, R.D. Testing the tail-dependent based on the radial component, Extremes, Volume 10 (2007), pp. 109-128 | MR | Zbl

[26] Falk, M.; Michel, R. Testing for Tail Independence in Extreme Value models, Annals of the Institute of Statistical Mathematics, Volume 58 (2006), pp. 261-290 | MR | Zbl

[27] Gardes, L.; Girard, S. Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique, Journal de la Société Française de Statistique, Volume 154 (2) (2013), pp. 98-118 | Numdam | MR | Zbl

[28] Guillou, A.; Naveau, P.; Schorgen, A. Madogram and asymptotic independence among maxima (2013) (Submitted) | MR

[29] Hüsler, J.; Deyuan, L. Testing asymptotic independence in bivariate extremes, Journal of Statistical Planning and Inference, Volume 139 (2009), pp. 990-998 | MR | Zbl

[30] Huser, R.; Davison, A. C. Space-time modelling of extreme events (2012) (Submitted) | MR

[31] Heffernan, J. E. A directory of coefficients of tail dependence, Extremes, Volume 3 (2000), pp. 279-290 | MR | Zbl

[32] Heffernan, J. E.; Tawn, J. A. A conditional approach for multivariate extreme values (with discussion), Journal of the Royal Statistical Society. Series B. Methodological, Volume 66 (2004), pp. 497-546 | MR | Zbl

[33] Joe, H. Parametric family of multivariate distributions with given margins, Journal of Multivariate Analysis, Volume 46 (1993), pp. 262-282 | MR | Zbl

[34] Joe, H. Multivariate models and dependence concepts, Monographs on Statistics and Applied Probability, 73, Chapman & Hall, London, 1997, xviii+399 pages | MR | Zbl

[35] Kabluchko, Z.; Schlather, M.; de Haan, L. Stationary max-stable fields associated to negative definite functions, The Annals of Probability, Volume 37 (2009), pp. 2042-2065 | MR | Zbl

[36] Lantuéjoul, C.; Bacro, J. N.; Bel, L. Storm processes and stochastic geometry, Extremes, Volume 14 (2011), pp. 413-428 | MR | Zbl

[37] Lindsay, B. G. Composite likelihood methods, Statistical inference from stochastic processes (Ithaca, NY, 1987) (Contemporary Mathematics), Volume 80, American Mathematical Society, Providence, RI, 1988, pp. 221-239 | MR | Zbl

[38] Ledford, A. W.; Tawn, J. A. Statistics for near independence in multivariate analysis, Biometrika, Volume 83 (1996), pp. 169-187 | MR | Zbl

[39] Ledford, A. W.; Tawn, J. A. Modelling dependence within joint tail regions, Journal of the Royal Statistical Society. Series B. Methodological, Volume 59 (1997), pp. 475-499 | MR | Zbl

[40] Naveau, P.; Guillou, A.; Cooley, D.; Diebolt, J. Modelling pairwise dependence of maxima in space, Biometrika, Volume 96 (2009) no. 1, pp. 1-17 | MR | Zbl

[41] Peng, L. Estimation of the coefficients of tail dependence in bivariate extremes, Statistics & Probability Letters, Volume 43 (1999), pp. 399-409 | MR | Zbl

[42] Pickands, J. Multivariate extreme value distributions, Proceedings of the 43rd session of the International Statistical Institute, Bulletin de l’Institut International de Statistique, Volume 49 (1981) no. 2, p. 859-878, 894-902 | MR | Zbl

[43] Padoan, S. A.; Ribatet, M.; Sisson, S.A. Likelihood-based inference for max-stable processes, Journal of the American Statistical Association, Volume 105 (2010), pp. 263-277 | MR | Zbl

[44] Poon, S.; Rockinger, M.; Tawn, J. A. Modelling extreme-value dependence in international stock markets, Statistica Sinica, Volume 13 (2003), pp. 929-953 | MR | Zbl

[45] Resnick, S.I. Extreme values, Regular variation and Point Processes, Springer, New York, 1987 | MR | Zbl

[46] Ribatet, M. Spatial extremes: Max-stable processes at work, Journal de la Société Française de Statistique, Volume 154 (2) (2013), pp. 156-177 | Numdam | MR | Zbl

[47] Ramos, A.; Ledford, A. W. Regular score tests of independence in multivariate extreme values, Extremes, Volume 8 (2005), pp. 5-26 | MR | Zbl

[48] Ramos, A.; Ledford, A. W. A new class of models for bivariate joint tails, Journal of the Royal Statistical Society. Series B. Methodological, Volume 71 (2009) no. 1, pp. 219-241 | MR | Zbl

[49] Ramos, A.; Ledford, A. W. An alternative point process framework for modeling multivariate extreme values, Communications in Statistics - Theory and Methods, Volume 40 (2011) no. 12, pp. 2205-2224 | MR | Zbl

[50] Schlather, M. Models for stationary max-stable random fields, Extremes, Volume 5 (2002), pp. 33-44 | MR | Zbl

[51] Sibuya, M. Bivariate extreme statistics, Annals of the Institute of Statistical Mathematics, Volume 11 (1960), pp. 195-210 | MR | Zbl

[52] Smith, R.L. Max-stable processes and spatial extremes (1990) (Technical report. University of Surrey)

[53] Schlather, M.; Tawn, J. A. Inequalities for the extremal coefficients of multivariate extreme value distributions, Extremes, Volume 51 (2002), pp. 87-102 | MR | Zbl

[54] Schlather, M.; Tawn, J. A. A dependence measure for multivariate and spatial extreme values: Properties and inference, Biometrika, Volume 90 (2003), pp. 139-156 | MR | Zbl

[55] Smith, R.L.; Tawn, J. A.; Coles, S.G. Markov chain models for threshold exceedances, Biometrika, Volume 84 (1997), pp. 249-268 | MR | Zbl

[56] Tawn, J. A. Bivariate extreme value theory: models and estimation, Biometrika, Volume 75 (1988), pp. 397-415 | MR | Zbl

[57] Varin, C. On composite marginal likelihoods, Advances in Statistical Analysis, Volume 92 (2008) no. 1, pp. 1-28 | MR | Zbl

[58] Varin, C.; Paolo, V. A note on composite likelihood inference and model selection, Biometrika, Volume 92 (2005) no. 3, pp. 519-528 | MR | Zbl

[59] Wadsworth, J. L.; Tawn, J. A. Dependence modelling for spatial extremes, Biometrika, Volume 99 (2012), pp. 253-272 | MR | Zbl

[60] Zhang, Z. Quotient correlation: A sample based alternative to Pearson’s correlation, The Annals of Statistics, Volume 36 (2008), pp. 1007-1030 | MR | Zbl

[61] Zhang, Z.; Qi, Y.; Ma, X. Asymptotic independence of correlation coefficients with application to testing hypothesis of independence, Electronic Journal of Statistics, Volume 5 (2011), pp. 342-372 | MR | Zbl