A hierarchical max-stable spatial model for extreme precipitation. (English) Zbl 1257.62120

Summary: Extreme environmental phenomena, such as major precipitation events, manifestly exhibit spatial dependence. Max-stable processes are a class of asymptotically-justified models that are capable of representing spatial dependence among extreme values. While these models satisfy modeling requirements, they are limited in their utility because their corresponding joint likelihoods are unknown for more than a trivial number of spatial locations, preventing, in particular, Bayesian analyses. We propose a new random effects model to account for spatial dependence. We show that our specification of the random effect distribution leads to a max-stable process that has the popular Gaussian extreme value process (GEVP) as a limiting case. The proposed model is used to analyze the yearly maximum precipitation from a regional climate model.


62P12 Applications of statistics to environmental and related topics
62M30 Inference from spatial processes
62G32 Statistics of extreme values; tail inference


spBayes; FRK
Full Text: DOI arXiv Euclid


[1] Banerjee, S., Carlin, B. P. and Gelfand, A. E. (2004). Hierarchical Modeling and Analysis for Spatial Data . Chapman & Hall/CRC, London. · Zbl 1053.62105
[2] Banerjee, S., Gelfand, A. E., Finley, A. O. and Sang, H. (2008). Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 825-848. · Zbl 05563371
[3] Blanchet, J. and Davison, A. C. (2011). Spatial modeling of extreme snow depth. Ann. Appl. Stat. 5 1699-1725. · Zbl 1228.62154
[4] Buishand, T. A., de Haan, L. and Zhou, C. (2008). On spatial extremes: With application to a rainfall problem. Ann. Appl. Stat. 2 624-642. · Zbl 1273.62258
[5] Coles, S. G. (1993). Regional modelling of extreme storms via max-stable processes. J. Roy. Statist. Soc. Ser. B 55 797-816. · Zbl 0781.60041
[6] Cooley, D., Naveau, P. and Poncet, P. (2006). Variograms for spatial max-stable random fields. In Dependence in Probability and Statistics. Lecture Notes in Statist. 187 373-390. Springer, New York. · Zbl 1110.62130
[7] Cooley, D., Nychka, D. and Naveau, P. (2007). Bayesian spatial modeling of extreme precipitation return levels. J. Amer. Statist. Assoc. 102 824-840. · Zbl 1469.62389
[8] Cressie, N. A. C. (1993). Statistics for Spatial Data . Wiley, New York. · Zbl 0799.62002
[9] Cressie, N. and Johannesson, G. (2008). Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. Ser. B Stat. Methodol. 70 209-226. · Zbl 05563351
[10] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab. 12 1194-1204. · Zbl 0597.60050
[11] de Haan, L. and Ferreira, A. (2006). Extreme Value Theory : An Introduction . Springer, New York. · Zbl 1101.62002
[12] Deheuvels, P. (1983). Point processes and multivariate extreme values. J. Multivariate Anal. 13 257-272. · Zbl 0519.60045
[13] Ehlert, A. and Schlather, M. (2008). Capturing the multivariate extremal index: Bounds and interconnections. Extremes 11 353-377. · Zbl 1199.60183
[14] Engelke, S., Kabluchko, Z. and Schlather, M. (2011). An equivalent representation of the Brown-Resnick process. Statist. Probab. Lett. 81 1150-1154. · Zbl 1234.60056
[15] Fougères, A.-L., Nolan, J. P. and Rootzén, H. (2009). Models for dependent extremes using stable mixtures. Scand. J. Stat. 36 42-59. · Zbl 1195.62067
[16] Fuentes, M., Henry, J. and Reich, B. J. (2012). Nonparametric spatial models for extremes: Applications to exterme temperature data. Extremes . · Zbl 1248.62170
[17] Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P., eds. (2010). Handbook of Spatial Statistics . CRC Press, Boca Raton, FL. · Zbl 1188.62284
[18] Genton, M. G., Ma, Y. and Sang, H. (2011). On the likelihood function of Gaussian max-stable processes. Biometrika 98 481-488. · Zbl 1215.62089
[19] Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55 231-244. · Zbl 0671.62007
[20] Higdon, D., Swall, J. and Kern, J. (1999). Non-stationary spatial modeling. In Bayesian Statistics 6 -Proceedings of the Sixth Valencia Meeting (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.) 761-768. Clarendon Press, Oxford. · Zbl 0982.62079
[21] Huser, R. and Davison, A. C. (2012). Space-time modelling of extreme events. Unpublished manuscript.
[22] Kabluchko, Z., Schlather, M. and de Haan, L. (2009). Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37 2042-2065. · Zbl 1208.60051
[23] Nakicenovic, N. et al. (2000). Special Report on Emissions Scenarios. A Special Report of Working Group III of the Intergovernmental Panel on Climate Change . Cambridge Univ. Press, Cambridge.
[24] Oesting, M., Kabluchko, Z. and Schlather, M. (2012). Simulation of Brown-Resnick processes. Extremes 15 89-107. · Zbl 1329.60157
[25] Padoan, S. A., Ribatet, M. and Sisson, S. A. (2010). Likelihood-based inference for max-stable processes. J. Amer. Statist. Assoc. 105 263-277. · Zbl 1397.62172
[26] Pickands, J. (1981). Multivariate extreme value distributions. In Proceedings 43 rd Session International Statistical Institute , Vol. 2 ( Buenos Aires , 1981). Bull. Inst. Internat. Statist. 49 859-878, 894-902. · Zbl 0518.62045
[27] Reich, B. J. and Fuentes, M. (2012). Nonparametric Bayesian models for a spatial covariance. Stat. Methodol. 9 265-274. · Zbl 1248.62170
[28] Reich, B. J., Fuentes, M., Herring, A. H. and Evenson, K. R. (2010). Bayesian variable selection for multivariate spatially varying coefficient regression. Biometrics 66 772-782. · Zbl 1203.62198
[29] Resnick, S. I. (1987). Extreme Values , Regular Variation , and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4 . Springer, New York. · Zbl 0633.60001
[30] Ribatet, M., Cooley, D. and Davison, A. (2012). Bayesian inference from composite likelihoods, with an application to spatial extremes. Statist. Sinica 22 813-845. · Zbl 1238.62031
[31] Sang, H. and Gelfand, A. E. (2010). Continuous spatial process models for spatial extreme values. J. Agric. Biol. Environ. Stat. 15 49-65. · Zbl 1306.62334
[32] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5 33-44. · Zbl 1035.60054
[33] Schlather, M. and Tawn, J. (2002). Inequalities for the extremal coefficients of multivariate extreme value distributions. Extremes 5 87-102. · Zbl 1035.60013
[34] Shaby, B. (2012). The open-faced sandwich adjustment for MCMC using estimating functions. Unpublished manuscript.
[35] Smith, R. L. (1990). Max-stable processes and spatial extremes. Unpublished manuscript.
[36] Smith, E. L. and Stephenson, A. G. (2009). An extended Gaussian max-stable process model for spatial extremes. J. Statist. Plann. Inference 139 1266-1275. · Zbl 1153.62067
[37] Stephenson, A. G. (2009). High-dimensional parametric modelling of multivariate extreme events. Aust. N. Z. J. Stat. 51 77-88. · Zbl 1336.62134
[38] Tawn, J. A. (1990). Modelling multivariate extreme value distributions. Biometrika 77 245-253. · Zbl 0716.62051
[39] Wang, Y. and Stoev, S. A. (2010a). Conditional sampling for max-stable random fields. Available at . 1005.0312
[40] Wang, Y. and Stoev, S. A. (2010b). On the structure and representations of max-stable processes. Adv. in Appl. Probab. 42 855-877. · Zbl 1210.60053
[41] Zhang, Z. and Smith, R. L. (2010). On the estimation and application of max-stable processes. J. Statist. Plann. Inference 140 1135-1153. · Zbl 1181.62150
[42] Zheng, Y., Zhu, J. and Roy, A. (2010). Nonparametric Bayesian inference for the spectral density function of a random field. Biometrika 97 238-245. · Zbl 1182.62070
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