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Characterization of multivariate heavy-tailed distribution families via copula. (English) Zbl 1236.62048
Summary: The multivariate regular variation (MRV) is one of the most important tools in modeling multivariate heavy-tailed phenomena. This paper characterizes the MRV distributions through the tail dependence function of the copula associated with them. Along with some existing results, our studies indicate that the existence of the lower tail dependence function of the survival copula is necessary and sufficient for a random vector with regularly varying univariate marginals to have a MRV tail. Moreover, the limit measure of the MRV tail is explicitly characterized. Our analysis is also extended to some more general multivariate heavy-tailed distributions, including the subexponential and the long-tailed distribution families.
MSC:
62H05Characterization and structure theory (Multivariate analysis)
62G32Statistics of extreme values; tail inference
References:
[1]Bingham, N.; Goldie, C.; Teugels, J.: Regular variation, (1987)
[2]Charpentier, A.; Segers, J.: Tails of multivariate Archimedean copulas, Journal of multivariate analysis 100, 1521-1537 (2009) · Zbl 1165.62038 · doi:10.1016/j.jmva.2008.12.015
[3]Cline, D. B. H.; Resnick, S. I.: Multivariate subexponential distributions, Stochastic process and their applications 42, 49-72 (1992) · Zbl 0751.62025 · doi:10.1016/0304-4149(92)90026-M
[4]Cline, D. B. H.; Samorodnitsky, G.: Subexponentiality of the product of independent random variables, Stochastic processes and their applications 49, 75-98 (1994) · Zbl 0799.60015 · doi:10.1016/0304-4149(94)90113-9
[5]De Haan, L.; Ferreira, A.: Extreme value theory: an introduction, (2006)
[6]De Haan, L.; Resnick, S. I.: Limit theory for multivariate sample extremes, Wahrscheinlichkeitstheorie und verwandte gebiete 40, 317-337 (1977) · Zbl 0375.60031 · doi:10.1007/BF00533086
[7]Embrechts, P.; Lindskog, F.; Mcneil, A.: Modelling dependence with copulas and applications to risk management, Handbook of heavy tailed distributions in finance, 329-384 (2003)
[8]Embrechts, P.; Klüppelberg, C.; Mikosch, T.: Modelling extremal events for insurance and finance, (1997)
[9]Finkenstädt, B.; Rootzén, H.: Extreme values in finance, telecommunications, and the environment, (2003)
[10]Joe, H.: Multivariate models and dependence concepts, (1997) · Zbl 0990.62517
[11]Kortschak, D.; Albrecher, H.: Asymptotic results for the sum of dependent non-identically distributed random variables, Methodology and computing in applied probability 11, 279-306 (2009) · Zbl 1171.60348 · doi:10.1007/s11009-007-9053-3
[12]Larsson, M.; Nešlehová, J.: Extremal behavior of Archimedean copulas, Advances in applied probability 43, 195-216 (2011) · Zbl 1213.62084 · doi:10.1239/aap/1300198519
[13]Li, H.; Sun, Y.: Tail dependence for heavy-tailed scale mixtures of multivariate distributions, Journal of applied probability 46, 925-937 (2009) · Zbl 1179.62076 · doi:10.1239/jap/1261670680
[14]Maulik, K.; Resnick, S. I.; Rootzén, H.: Asymptotic independence and a network traffic model, Journal of applied probability 39, 671-699 (2002) · Zbl 1090.90017 · doi:10.1239/jap/1037816012
[15]Mcneil, A. J.; Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions, Annals of statistics 37, 3059-3097 (2009) · Zbl 1173.62044 · doi:10.1214/07-AOS556
[16]Nelsen, R. B.: An introduction to copulas, (2006)
[17]Rachev, S. T.: Handbook of heavy tailed distributions in finance, (2003)
[18]Resnick, S. I.: Extreme values, regular variation and point processes, (1987)
[19]Resnick, S. I.: Heavy-tail phenomena: probabilistic and statistical modeling, (2007)
[20]Y. Sun, H. Li, Tail approximation of Value-at-Risk under multivariate regular variation, Unpublished results, 2010, Accessible on Nov 2nd, 2011 through the link http://www.math.wsu.edu/math/faculty/lih/VaR-Sun_Li-Nov10.pdf.