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

German Bernhart; Jan-Frederik Mai; Matthias Scherer

Dependence Modeling (2015)

  • Volume: 3, Issue: 1, page 29-46, electronic only
  • ISSN: 2300-2298

Abstract

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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.

How to cite

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German Bernhart, Jan-Frederik Mai, and Matthias Scherer. "On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions." Dependence Modeling 3.1 (2015): 29-46, electronic only. <http://eudml.org/doc/271008>.

@article{GermanBernhart2015,
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.},
author = {German Bernhart, Jan-Frederik Mai, Matthias Scherer},
journal = {Dependence Modeling},
keywords = {MSMVE distributions; Bernstein functions; IDT-frailty copulas; IDT processes; extreme-value copulas},
language = {eng},
number = {1},
pages = {29-46, electronic only},
title = {On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions},
url = {http://eudml.org/doc/271008},
volume = {3},
year = {2015},
}

TY - JOUR
AU - German Bernhart
AU - Jan-Frederik Mai
AU - Matthias Scherer
TI - On the construction of low-parametric families of min-stable multivariate exponential distributions in large dimensions
JO - Dependence Modeling
PY - 2015
VL - 3
IS - 1
SP - 29
EP - 46, electronic only
AB - 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.
LA - eng
KW - MSMVE distributions; Bernstein functions; IDT-frailty copulas; IDT processes; extreme-value copulas
UR - http://eudml.org/doc/271008
ER -

References

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  1. [1] Ballani, F. and Schlather, M. (2011). A construction principle for multivariate extreme value distributions. Biometrika, 98(3):633-645. [Crossref][WoS] Zbl1230.62073
  2. [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. Zbl1111.60028
  3. [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. Zbl1102.60013
  4. [4] Barndorff-Nielsen, O. E., Rosinski, J., and Thorbjornsen, S. (2008). General Y-transformations. Alea, 4:131-165. 
  5. [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. 
  6. [6] Cherubini, U., Luciano, E., and Vecchiato, W. (2004). Copula Methods in Finance. John Wiley & Sons, Chichester. Zbl1163.62081
  7. [7] De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab., 12(4):1194-1204. [Crossref] Zbl0597.60050
  8. [8] De Haan, L. and Pickands, J. (1986). Stationary min-stable stochastic processes. Probab. Theory Rel. Fields, 72(4):477-492. [Crossref] Zbl0577.60034
  9. [9] De Haan, L. and Resnick, S. (1977). Limit theory for multivariate sample extremes. Z. Wahrsch. verw. Gebiete, 40(4):317-337. Zbl0375.60031
  10. [10] Durante, F. and Salvadori, G. (2010). On the construction of multivariate extreme value models via copulas. Environmetrics, 21(2):143-161. 
  11. [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. [WoS] Zbl1216.60042
  12. [12] Esary, J. D. and Marshall, A. W. (1974). Multivariate distributions with exponential minimums. Ann. Statist., 2:84-98. [Crossref] Zbl0293.60017
  13. [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. Zbl1195.62067
  14. [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. Zbl06085266
  15. [15] Gumbel, E. J. and Goldstein, N. (1964). Analysis of empirical bivariate extremal distributions. J. Amer. Statist. Assoc., 59(307):794-816. Zbl0129.11404
  16. [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. 
  17. [17] Hürlimann,W. (2003). Hutchinson-Lai’s conjecture for bivariate extreme value copulas. Statist. Probab. Lett., 61(2):191-198. Zbl1101.62340
  18. [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. [WoS][Crossref] Zbl0998.62050
  19. [19] Joe, H. (1990). Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Lett., 9(1):75-81. Zbl0686.62035
  20. [20] Joe, H. (1997). Multivariate Models and Multivariate Dependence Concepts. Chapman & Hall/CRC. Zbl0990.62517
  21. [21] Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall/CRC. Zbl06345338
  22. [22] Jurek, Z. J. (1985). Relations between the s-selfdecomposable and selfdecomposable measures. Ann. Probab., 13(2):592- 608. [Crossref] Zbl0569.60011
  23. [23] Klenke, A. (2006). Wahrscheinlichkeitstheorie. Springer, Berlin. 
  24. [24] Kotz, S. and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications. Imperial College Press, London. Zbl0960.62051
  25. [25] Longin, F. and Solnik, B. (2001). Extreme correlation of international equity markets. J. Finance, 56(2):649-676. 
  26. [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. 
  27. [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. Zbl1310.62072
  28. [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. Zbl1273.62070
  29. [29] Mansuy, R. (2005). On processes which are infinitely divisible with respect to time. Working paper, arxiv.org/abs/math/ 0504408. 
  30. [30] Molchanov, I. (2008). Convex geometry of max-stable distributions. Extremes, 11(3):235-259. Zbl1164.60003
  31. [31] Nelsen, R. B. (2006). An Introduction to Copulas. Springer, New York. Zbl1152.62030
  32. [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. Zbl0672.62065
  33. [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. [Crossref] 
  34. [34] Rajput, B. S. and Rosinski, J. (1989). Spectral representations of infinitely divisible processes. Probab. Theory Rel. Fields, 82(3):451-487. [Crossref] Zbl0659.60078
  35. [35] Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York. Zbl0633.60001
  36. [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. [Crossref][WoS] Zbl1283.60021
  37. [37] Sato, K.-I. (1999). Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge. Zbl0973.60001
  38. [38] Sato, K.-I. (2004). Stochastic integrals in additive processes and application to semi-Lévy processes. Osaka J. Math., 41(1):211-236. Zbl1050.60054
  39. [39] Schilling, R., Song, R., and Vondracek, Z. (2010). Bernstein Functions. De Gruyter, Berlin. Zbl1197.33002
  40. [40] Schönbucher, P. J. and Schubert, D. (2001). Copula-dependent defaults in intensity models. Working paper, http://ssrn. com/abstract=301968. 
  41. [41] Segers, J. (2012). Max-stable models for multivariate extremes. REVSTAT, 10(1):61-82. Zbl1297.62121
  42. [42] Vasicek, O. A. (2002). Loan portfolio value. Risk, 160-162. 
  43. [43] Williamson, R. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J., 23(2):189-207. [Crossref] Zbl0070.28501

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