Abstract
In this article we discuss the asymptotic behaviour of the componentwise maxima for a specific bivariate triangular array. Its components are given in terms of linear transformations of bivariate generalised symmetrised Dirichlet random vectors introduced in Fang and Fang (Statistical inference in elliptically contoured and related distributions. Allerton Press, New York, 1990). We show that the componentwise maxima of such triangular arrays is attracted by a bivariate max-infinitely divisible distribution function, provided that the associated random radius is in the Weibull max-domain of attraction.
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References
Berman MS (1992) Sojourns and extremes of stochastic processes. Wadsworth & Brooks Cole
Cambanis S, Huang S and Simons G (1981). On the theory of elliptically contoured distributions. J Multivariate Anal 11(3): 368–385
De Haan L and Ferreira A (2006). Extreme value theory. An introduction. Springer, Heidelberg
Eddy WF and Gale JD (1981). The convex hull of a spherically symmetric sample. Adv Appl Prob 13: 751–763
Falk M, Hüsler J, Reiss R-D (2004) Laws of small numbers: extremes and rare events. DMV seminar, vol 23, 2nd edn. Birkhäuser, Basel
Fang K-T and Fang Bi-Qi (1990). Generalised symmetrised Dirichlet distributions. In: Fang, KT and Anderson, TW (eds) Statistical inference in elliptically contoured and related distributions, pp 127–136. Allerton Press, New York
Fang K-T, Kotz S and Ng K-W (1990). Symmetric multivariate and related distributions. Chapman & Hall, London
Gale JD (1980) The asymptotic distribution of the convex hull of a random sample. PhD Thesis, Carnegie-Mellon University
Hashorva E (2005a). Elliptical triangular arrays in the max-domain of attraction of Hüsler–Reiss distributon. Stat Prob Lett 72(2): 125–135
Hashorva E (2005b). Extremes of asymptotically spherical and elliptical random vectors. Insurance Math Econ 36(3): 285–302
Hashorva E (2006a). On the max-domain of attractions of bivariate elliptical arrays. Extremes 8(3): 225–233
Hashorva E (2006b). A novel class of bivariate max-stable distributions. Stat Prob Lett 76(10): 1047–1055
Hashorva E (2006c). On the multivariate Hüsler–Reiss distribution attracting the maxima of elliptical triangular arrays. Stat Prob Lett 76(18): 2027–2035
Hashorva E, Kotz S and Kume A (2007). L p -norm generalised symmetrised Dirichlet distributions. Albanian J Math 1(1): 31–56
Hüsler J and Reiss R-D (1989). Maxima of normal random vectors: between independence and complete dependence. Stat Prob Lett 7: 283–286
Kallenberg O (1997). Foundations of modern probability. Springer, New York
Kotz S, Balakrishnan N and Johnson NL (2000). Continuous multivariate distributions, 2nd edn. Wiley, New York
Reiss R-D (1989). Approximate distributions of order statistics: with applications to nonparametric statistics. Springer, New York
Resnick SI (1987). Extreme values, regular variation and point processes. Springer, New York
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Hashorva, E. A new family of bivariate max-infinitely divisible distributions. Metrika 68, 289–304 (2008). https://doi.org/10.1007/s00184-007-0158-z
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DOI: https://doi.org/10.1007/s00184-007-0158-z