×

Hierarchical Archimedean copulas through multivariate compound distributions. (English) Zbl 1395.62112

Summary: In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E05 Probability distributions: general theory
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

CopulaModel
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bedford, T.; Cooke, R. M., Vines: A new graphical model for dependent random variables, Ann. Statist., 1031-1068 (2002) · Zbl 1101.62339
[2] Brechmann, E. C., Hierarchical kendall copulas: Properties and inference, Canad. J. Statist., 42, 1, 78-108 (2014) · Zbl 1349.62172
[3] Cont, R.; Tankov, P., Financial Modelling with Jump Processes, vol. 2, 2003 (2003), Chapman & Hall/CRC, London
[5] Devroye, L., Random variate generation for exponentially and polynomially tilted stable distributions, ACM Trans Modeling Comput. Simul., 19, 4, 18 (2009) · Zbl 1390.65008
[6] Feller, W., An Introduction to Probability and Its Applications, Vol. II (1971), Wiley: Wiley New York · Zbl 0219.60003
[7] Górecki, J.; Hofert, M.; Holeňa, M., An approach to structure determination and estimation of hierarchical archimedean copulas and its application to bayesian classification, J. Intell. Inf. Syst., 46, 1, 21-59 (2016)
[9] Górecki, J.; Holeňa, M., Structure determination and estimation of hierarchical archimedean copulas based on kendall correlation matrix, (International Workshop on New Frontiers in Mining Complex Patterns (2013), Springer), 132-147
[10] Hering, C.; Hofert, M.; Mai, J.-F.; Scherer, M., Constructing hierarchical archimedean copulas with lévy subordinators, J. Multivariate Anal., 101, 6, 1428-1433 (2010) · Zbl 1194.60017
[11] Hofert, M., Sampling Nested Archimedean Copulas with Applications to cdo Pricing (2010), Universität Ulm, (Ph.D. Thesis)
[12] Hofert, M., Efficiently sampling nested Archimedean copulas, Comput. Statist. Data Anal., 55, 1, 57-70 (2011) · Zbl 1247.62132
[13] Hofert, M., A stochastic representation and sampling algorithm for nested archimedean copulas, J. Stat. Comput. Simul., 82, 9, 1239-1255 (2012) · Zbl 1271.60026
[14] Joe, H., Multivariate Models and Multivariate Dependence Concepts (1997), CRC Press · Zbl 0990.62517
[15] Joe, H., Dependence Modeling with Copulas (2014), CRC Press · Zbl 1346.62001
[16] Kimberling, C. H., A probabilistic interpretation of complete monotonicity, Aequationes Math., 10, 2, 152-164 (1974) · Zbl 0309.60012
[17] Marshall, A. W.; Olkin, I., Families of multivariate distributions, J. Amer. Statist. Assoc., 83, 403, 834-841 (1988) · Zbl 0683.62029
[18] McNeil, A. J., Sampling nested archimedean copulas, J. Stat. Comput. Simul., 78, 6, 567-581 (2008) · Zbl 1221.00061
[19] Nelsen, R. B., An Introduction to Copulas (2007), Springer Science & Business Media
[20] Okhrin, O.; Okhrin, Y.; Schmid, W., On the structure and estimation of hierarchical archimedean copulas, J. Econometrics, 173, 2, 189-204 (2013) · Zbl 1443.62137
[21] Segers, J.; Uyttendaele, N., Nonparametric estimation of the tree structure of a nested archimedean copula, Comput. Statist. Data Anal., 72, 190-204 (2014) · Zbl 1506.62163
[22] Sklar, M., Fonctions de répartition à n dimensions et leurs marges (1959), Université Paris 8 · Zbl 0100.14202
[24] Zhu, W.; Wang, C.-W.; Tan, K. S., Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests, J. Bank. Finance, 69, 20-36 (2016)
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.