Abstract
The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.
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References
Barndorff-Nielsen, O.E. and Cox, D.R., Asymptotic Techniques for Use in Statistics, Chapman & Hall, London, 1989.
Coles, S.G., “Regional modeling of extreme storms via max-stable processes,” J. R. Statist. Soc. B. 55, 797–816, (1993).
Coles, S.G., Heffernan, J.E., and Tawn, J.A., “Dependence measures for extreme value analyses,” Extremes 2, 339–365, (1999).
Coles, S.G. and Tawn, J.A., “Modelling extremes of the areal rainfall process,” J. R. Statist. Soc. B. 58, 329–347, (1996).
Deheuvels, P., “Point processes and multivariate extreme values,” J. Multivar. Anal. 13, 257–272, (1983).
Hall, P. and Tajvidi, N., “Distribution and dependence-function estimation for bivariate extreme-value distributions,” Bernoulli 6, 835–844, (2000).
Pickands, J., “Multivariate extreme value distributions,” Bull. Int. Stat. Inst. 49, 859–878, (1981).
Resnick, S.I., Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987.
Schlather, M. and Tawn, J.A., Properties of extremal coefficients. Technical Report ST–00–01, Lancaster University, Lancaster, 2000.
Schlather, M. and Tawn, J.A., “A dependence measure for multivariate and spatial extreme values: Properties and inference,” Biometrika, 2002, to appear.
Smith, R.L., Max-stable processes and spatial extremes. Unpublished manuscript, 1990.
Smith, R.L., Tawn, J.A., and Yuen, H.-K., “Statistics of multivariate extremes,” Int. Statist. Rev. 58, 47–58, (1990).
Tiago de Oliveira, J., “Structure theory of bivariate extremes, extensions,” Est. Mat., Estat. e. Econ. 7, 165–195, (1962/63).
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Schlather, M., Tawn, J. Inequalities for the Extremal Coefficients of Multivariate Extreme Value Distributions. Extremes 5, 87–102 (2002). https://doi.org/10.1023/A:1020938210765
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DOI: https://doi.org/10.1023/A:1020938210765