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Dependence Modeling

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On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions

German Bernhart / Jan-Frederik Mai / Matthias Scherer
Published Online: 2015-05-22 | DOI: https://doi.org/10.1515/demo-2015-0003

Abstract

Min-stable multivariate exponential (MSMVE) distributions constitute an important family of distributions, among others due to their relation to extreme-value distributions. Being true multivariate exponential models, they also represent a natural choicewhen modeling default times in credit portfolios. Despite being well-studied on an abstract level, the number of known parametric families is small. Furthermore, for most families only implicit stochastic representations are known. The present paper develops new parametric families of MSMVE distributions in arbitrary dimensions. Furthermore, a convenient stochastic representation is stated for such models, which is helpful with regard to sampling strategies.

Keywords: MSMVE distributions; Bernstein functions; IDT-frailty copulas; IDT processes; extreme-value copulas

MSC:: 60G70

References

  • [1] Ballani, F. and Schlather, M. (2011). A construction principle for multivariate extreme value distributions. Biometrika, 98(3):633-645. CrossrefWeb of ScienceGoogle Scholar

  • [2] Barndorff-Nielsen, O. E.,Maejima, M., and Sato, K.-I. (2006a). Infinite divisibility for stochastic processes and time change. J. Theoret. Probab., 19(2):411-446. Google Scholar

  • [3] Barndorff-Nielsen, O. E.,Maejima, M., and Sato, K.-I. (2006b). Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli, 12(1):1-33. Google Scholar

  • [4] Barndorff-Nielsen, O. E., Rosinski, J., and Thorbjornsen, S. (2008). General Y-transformations. Alea, 4:131-165. Google Scholar

  • [5] Brigo, D. and Chourdakis, K. (2012). Consistent single- and multi-step sampling of multivariate arrival times: A characterization of self-chaining copulas. Working paper, available at arxiv.org/abs/1204.2090. Google Scholar

  • [6] Cherubini, U., Luciano, E., and Vecchiato, W. (2004). Copula Methods in Finance. John Wiley & Sons, Chichester. Google Scholar

  • [7] De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab., 12(4):1194-1204. CrossrefGoogle Scholar

  • [8] De Haan, L. and Pickands, J. (1986). Stationary min-stable stochastic processes. Probab. Theory Rel. Fields, 72(4):477-492. CrossrefGoogle Scholar

  • [9] De Haan, L. and Resnick, S. (1977). Limit theory for multivariate sample extremes. Z. Wahrsch. verw. Gebiete, 40(4):317-337. Google Scholar

  • [10] Durante, F. and Salvadori, G. (2010). On the construction of multivariate extreme value models via copulas. Environmetrics, 21(2):143-161. Google Scholar

  • [11] Es-Sebaiy, K. and Ouknine, Y. (2007). How rich is the class of processes which are infinitely divisible with respect to time? Statist. Probab. Lett., 78(5):537-547. Web of ScienceGoogle Scholar

  • [12] Esary, J. D. and Marshall, A. W. (1974). Multivariate distributions with exponential minimums. Ann. Statist., 2:84-98. CrossrefGoogle Scholar

  • [13] Fougères, A.-L., Nolan, J. P., and Rootzén, H. (2009). Models for dependent extremes using stable mixtures. Scand. J. Stat., 36(1):42-59. Google Scholar

  • [14] Gudendorf, G. and Segers, J. (2010). Extreme-value copulas. In Jaworski, P., Durante, F., Härdle,W. K., and Rychlik, T., editors, Copula Theory and its Applications, 127-145. Springer, Berlin. Google Scholar

  • [15] Gumbel, E. J. and Goldstein, N. (1964). Analysis of empirical bivariate extremal distributions. J. Amer. Statist. Assoc., 59(307):794-816. Google Scholar

  • [16] Hofmann, D. (2009). Characterization of the D-Norm Corresponding to aMultivariate Extreme Value Distribution. PhD thesis, Universität Würzburg, http://opus.bibliothek.uni-wuerzburg.de/frontdoor/index/index/docId/3454. Google Scholar

  • [17] Hürlimann,W. (2003). Hutchinson-Lai’s conjecture for bivariate extreme value copulas. Statist. Probab. Lett., 61(2):191-198. Google Scholar

  • [18] Jiménez, J. R., Villa-Diharce, E., and Flores, M. (2001). Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Multivariate Anal., 76(2):159-191. Web of ScienceCrossrefGoogle Scholar

  • [19] Joe, H. (1990). Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Lett., 9(1):75-81. Google Scholar

  • [20] Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Chapman & Hall/CRC. Google Scholar

  • [21] Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall/CRC. Google Scholar

  • [22] Jurek, Z. J. (1985). Relations between the s-selfdecomposable and selfdecomposable measures. Ann. Probab., 13(2):592- 608. CrossrefGoogle Scholar

  • [23] Klenke, A. (2006). Wahrscheinlichkeitstheorie. Springer, Berlin. Google Scholar

  • [24] Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications. Imperial College Press, London. Google Scholar

  • [25] Longin, F. and Solnik, B. (2001). Extreme correlation of international equity markets. J. Finance, 56(2):649-676. Google Scholar

  • [26] Mai, J.-F. (2014). Mutivariate exponential distributions with latent factor structure and related topics. Habilitation Thesis, Technische Universität München, https://mediatum.ub.tum.de/node?id=1236170. Google Scholar

  • [27] Mai, J.-F. and Scherer, M. (2014). Characterization of extendible distributions with exponential minima via processes that are infinitely divisible with respect to time. Extremes, 17(1):77-95. Google Scholar

  • [28] Mai, J.-F., Scherer, M., and Zagst, R. (2013). CIID frailty models and implied copulas. In Jaworski, P., Durante, F., and Härdle, W. K., editors, Copulae in Mathematical and Quantitative Finance, 201-230. Springer, Berlin. Google Scholar

  • [29] Mansuy, R. (2005). On processes which are infinitely divisible with respect to time. Working paper, arxiv.org/abs/math/ 0504408. Google Scholar

  • [30] Molchanov, I. (2008). Convex geometry of max-stable distributions. Extremes, 11(3):235-259. Google Scholar

  • [31] Nelsen, R. B. (2006). An Introduction to Copulas. Springer, New York. Google Scholar

  • [32] Pickands, J. (1989).Multivariate negative exponential and extreme value distributions. In Hüsler, J. and Reiss, R.-D., editors, Extreme Value Theory, 262-274. Springer, New York. Google Scholar

  • [33] Poon, S.-H., Rockinger, M., and Tawn, J. (2004). Extreme value dependence in financial markets: Diagnostics, models, and financial implications. Rev. Financ. Stud., 17(2):581-610. CrossrefGoogle Scholar

  • [34] Rajput, B. S. and Rosinski, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Rel. Fields, 82(3):451-487. CrossrefGoogle Scholar

  • [35] Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York. Google Scholar

  • [36] Ressel, P. (2013). Homogeneous distributions - and a spectral representation of classical mean values and stable tail dependence functions. J. Multivariate Anal., 117:246-256. CrossrefWeb of ScienceGoogle Scholar

  • [37] Sato, K.-I. (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge. Google Scholar

  • [38] Sato, K.-I. (2004). Stochastic integrals in additive processes and application to semi-Lévy processes. Osaka J. Math., 41(1):211-236. Google Scholar

  • [39] Schilling, R., Song, R., and Vondracek, Z. (2010). Bernstein Functions. De Gruyter, Berlin. Google Scholar

  • [40] Schönbucher, P. J. and Schubert, D. (2001). Copula-dependent defaults in intensity models. Working paper, http://ssrn. com/abstract=301968. Google Scholar

  • [41] Segers, J. (2012). Max-stable models for multivariate extremes. REVSTAT, 10(1):61-82. Google Scholar

  • [42] Vasicek, O. A. (2002). Loan portfolio value. Risk, 160-162. Google Scholar

  • [43] Williamson, R. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J., 23(2):189-207. CrossrefGoogle Scholar

About the article


Received: 2015-01-13

Accepted: 2015-05-07

Published Online: 2015-05-22


Citation Information: Dependence Modeling, Volume 3, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2015-0003.

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© 2015 German Bernhart et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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