Abstract
Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.
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References
Ancona-Navarrete, M.A. and Tawn, J.A., “Diagnostics for extremal dependence in spatial processes,” Submitted to Extremes, 2002.
Barndorff-Nielsen, O.E., Gupta, V.K., Pérez-Abreu, V., and Waymire, E., (eds), Stochastic Methods in Hydrology, World Scientific, Singapore, New Jersey, London, Hong Kong, 1998.
Casson, E. and Coles, S.G., “Spatial regression models for extremes,” Extremes 1, 449–468, (1999).
Casson, E. and Coles, S.G., “Simulation and extremal analysis of hurricane events,” J. R. Statist. Soc., Ser. C 49, 227–245, (2000).
Coles, S.G., “Regional modelling of extreme storms via max-stable processes,” J. R. Statist. Soc., Ser. B. 55, 797–816, (1993).
Coles, S.G. and Tawn, J.A., “Modelling extremes of the areal rainfall process,” J. R. Statist. Soc., Ser. B 58, 329–347, (1996).
Davis, R.A. and Resnick, S.I., “Prediction of stationary max-stable processes,” Ann. Appl. Probab. 3, 497–525, (1993).
de Haan, L., “A spectral representation for max-stable processes,” Ann. Probab. 12, 1194–1204, (1984).
de Haan, L. and Pickands, J., “Stationary min-stable stochastic processes,” Probab. Th. Rel. Fields. 72, 477–492, (1986).
Deheuvels, P., “Point processes and multivariate extreme values,” J. Multivar. Anal. 13, 257–272, (1983).
Giné, E., Hahn, M.G., and Vatan, P., “Max-infinitely divisible and max-stable sample continuous processes,” Probab. Th. Rel. Fields 87, 139–165, (1990).
Heinrich, L. and Molchanov, I.S., “Some limit theorems for extremal and union shot-noise processes,” Math. Nachr. 168, 139–159, (1994).
Ihaka, R. and Gentleman, R., “R: A language for data analysis and graphics,” J. Comput. Graph. Stat. 5(3), 299–314, (1996).
Jeulin, D. and Jeulin, P., “Synthesis of rough surfaces of random morphological functions,” Stereo. Iugosl. 3, 239–246, (1981).
Matheron, G., Random Sets and Integral Geometry, John Wiley & Sons, New York, 1975.
Norberg, T., On the existence and convergence of probabiity measures on continuous semi-lattices. Technical Report 148, Center for Stochastic Processes, University of North Carolina, 1986.
Pickands, J., “Multivariate extreme value distributions,” Bull. Int. Stat. Inst. 49, 859–878, (1981).
Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, Volume 4, Springer, New York, Berlin, 1987.
Schlather, M., “Simulation and analysis of random fields,” R News 1(2), 18–20, (2001).
Smith, R.L., “Max-stable processes and spatial extremes,” Unpublished manuscript, 1990.
Stoyan, D., Kendall, W.S., and Mecke, J., Stochastic Geometry and its Applications (second edition), John Wiley & Sons, Chichester, 1995.
Tawn, J.A., “An extreme-value theory model for dependent observations,” J. Hydrology 101, 227–250, (1988).
Wackernagel, H., Multivariate Geostatistics (second edition), Springer, Berlin, Heidelberg, 1998.
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Schlather, M. Models for Stationary Max-Stable Random Fields. Extremes 5, 33–44 (2002). https://doi.org/10.1023/A:1020977924878
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- bivariate extreme value distribution
- dependence function
- Gaussian random field
- max-stable random field
- rainfall modeling
- simulation of max-stable processes