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Bivariate distributions with given extreme value attractor. (English) Zbl 0978.62043
Summary: A new class of bivariate distributions is introduced and studied, which encompasses Archimedean copulas and extreme value distributions as special cases. Its dependence structure is described, its maximum and minimum attractors are determined, and an algorithm is given for generating observations from any member of this class. It is also shown how it is possible to construct distributions in this family with a predetermined extreme value attractor.
This construction is used to study via simulation the small-sample behavior of a bivariate threshold method suggested by H. Joe, R.L. Smith, and I. Weissman [J. R. Stat. Soc., Ser. B 54, No. 1, 171-183 (1992)] for estimating the joint distribution of extremes of two random variates.

MSC:
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G32 Statistics of extreme values; tail inference
60G70 Extreme value theory; extremal stochastic processes
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