×

Lévy-frailty copulas. (English) Zbl 1162.62048

Summary: A parametric family of \(n\)-dimensional extreme-value copulas of A.W. Marshall and I. Olkin type [J. Am. Stat. Assoc. 83, No. 403, 834–841 (1988; Zbl 0683.62029)] is introduced. Members of this class arise as survival copulas in Lévy-frailty models. The underlying probabilistic construction introduces dependence to initially independent exponential random variables by means of first-passage times of a Lévy subordinator. Jumps of the subordinator correspond to a singular component of the copula. Additionally, a characterization of completely monotone sequences via the introduced family of copulas is derived. An alternative characterization is given by Hausdorff’s moment problem in terms of random variables with compact support.
The resulting correspondence between random variables, Lévy subordinators, and copulas is studied and illustrated with several examples. Finally, it is used to provide a general methodology for sampling the copula in many cases. The new class is shown to share some properties with Archimedean copulas regarding construction and analytical form. Finally, the parametric form allows us to compute different measures of dependence and the J. Pickands [Bull. Int. Stat. Inst. 49, No. 2, 859–878 (1981; Zbl 0518.62045)] representation.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62H20 Measures of association (correlation, canonical correlation, etc.)
60G51 Processes with independent increments; Lévy processes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Joe, H., Multivariate Models and Dependence Concepts (1997), Chapman and Hall/CRC · Zbl 0990.62517
[2] Nelsen, R. B., An Introduction to Copulas (1998), Springer-Verlag
[3] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative Risk Management (2005), Princeton University Press · Zbl 1089.91037
[4] Sklar, A., Fonctions de répartition à n dimensions et leurs marges, Publications de L’Institut de Statistique de L’Université de Paris, 8, 229-231 (1959) · Zbl 0100.14202
[5] Embrechts, P.; Lindskog, F.; McNeil, A. J., (Rachev, S. T., Modelling Dependence with Copulas and Applications to Risk Management. Modelling Dependence with Copulas and Applications to Risk Management, Handbook of Heavy Tailed Distributions in Finance (2001), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam)
[6] Schmid, F.; Schmidt, R., Bootstrapping Spearman’s multivariate rho, Proc. Compstat., 6, 759-766 (2006)
[7] Schmid, F.; Schmidt, R., Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence, Metrika, 66, 3, 323-354 (2007) · Zbl 1433.62151
[8] Frahm, G., On the extremal dependence coefficient of multivariate distributions, Statistics & Probability Letters, 76, 1470-1481 (2006) · Zbl 1120.62035
[9] Cuadras, C. M.; Augé, J., A continuous general multivariate distribution and its properties, Communications in Statistics - Theory and Methods, 10, 4, 339-353 (1981) · Zbl 0456.62013
[10] J.-F. Mai, M. Scherer, Efficiently sampling exchangeable Cuadras-Augé copulas in high dimensions, Information Sciences (2008) (in press); J.-F. Mai, M. Scherer, Efficiently sampling exchangeable Cuadras-Augé copulas in high dimensions, Information Sciences (2008) (in press) · Zbl 1171.62036
[11] Falk, M.; Hüsler, J.; Reiss, R.-D., Laws of Small Numbers: Extremes and Rare Events (2004), Birkhäuser Verlag: Birkhäuser Verlag Basle, Boston, Berlin · Zbl 1083.60003
[12] Marshall, A. W.; Olkin, I., A multivariate exponential distribution, Journal of the American Statistical Association, 62, 317, 30-44 (1967) · Zbl 0147.38106
[13] Durante, F.; Quesada-Molina, J. J.; Úbeda-Flores, M., On a family of multivariate copulas for aggregation processes, Information Sciences, 177, 5715-5724 (2007) · Zbl 1132.68761
[14] J.-F. Mai, M. Scherer, A tractable multivariate default model based on a stochastic time-change, International Journal of Theoretical and Applied Finance (2008) (in press); J.-F. Mai, M. Scherer, A tractable multivariate default model based on a stochastic time-change, International Journal of Theoretical and Applied Finance (2008) (in press)
[15] Marshall, A. W.; Olkin, I., Families of multivariate distributions, Journal of the American Statistical Association, 83, 834-841 (1988) · Zbl 0683.62029
[16] Oakes, D., Bivariate survival models induced by frailties, Journal of the American Statistical Association, 84, 487-493 (1989) · Zbl 0677.62094
[17] A.J. McNeil, J. Nešlehová, Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions, Annals of Statistics (2007) (in press). Available at http://www.ma.hw.ac.uk/mcneil/ftp/McNeil-Neslehova-07.pdf; A.J. McNeil, J. Nešlehová, Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions, Annals of Statistics (2007) (in press). Available at http://www.ma.hw.ac.uk/mcneil/ftp/McNeil-Neslehova-07.pdf
[18] Kimberling, C. H., A probabilistic interpretation of complete monotonicity, Aequationes Mathematicae, 10, 152-164 (1974) · Zbl 0309.60012
[19] Pickands, J., Multivariate extreme value distributions, Bulletin of the International Statistical Institute: Proceedings of the 43rd Session (Buenos Aires), 859-878 (1981) · Zbl 0518.62045
[20] Widder, D. V., Laplace Transform (1942), Princeton University Press · Zbl 0060.24801
[21] Feller, W., An Introduction to Probability Theory and Its Applications, Vol. II (1966), John Wiley and Sons Inc. · Zbl 0138.10207
[22] Karlin, S., Total Positivity (1968), Stanford University Press · Zbl 0219.47030
[23] Lorch, L.; Newman, D. J., On the composition of completely monotonic functions and completely monotonic sequences and related questions, Journal of the London Mathematical Society, s2-28, 1, 31-45 (1983) · Zbl 0547.26010
[24] Gnedin, A.; Pitman, J., Moments of convex distribution functions and completely alternating sequences, (Probability and Statistics: Essays in Honor of David A. Freedman, vol. 2 (2008)), 30-41 · Zbl 1176.60027
[25] Bertoin, J., Lévy Processes (1996), Cambridge University Press · Zbl 0861.60003
[26] Sato, K.-I., Lévy Processes and Infinitely Divisible Distributions (1999), Cambridge University Press · Zbl 0973.60001
[27] Schoutens, W., (Lévy Processes in Finance: Pricing Financial Derivatives. Lévy Processes in Finance: Pricing Financial Derivatives, Wiley Series in Probability and Statistics (2003))
[28] Applebaum, D., Lévy Processes and Stochastic Calculus (2004), Cambridge University Press · Zbl 1073.60002
[29] Cont, R.; Tankov, P., Financial Modelling With Jump Processes, (Financial Mathematics Series (2004), Chapman and Hall/CRC) · Zbl 1052.91043
[30] Hausdorff, F., Summationsmethoden und Momentfolgen I, Mathematische Zeitschrift, I, 74-109 (1921) · JFM 48.2005.01
[31] Müntz, C. H., über den Approximationssatz von Weierstrass, Festschrift H. A. Schwarz, 303-312 (1914) · JFM 45.0633.02
[32] Szász, O., Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen, Mathematische Annalen, 77, 482-496 (1916) · JFM 46.0419.03
[33] Gupta, A. K.; Nadarajah, S., Handbook of Beta Distribution and Its Applications (2004), Marcel Dekker · Zbl 1062.62021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.