Geodesic
A short note on multivariate dependence modeling
Kybernetika, Tome 49 (2013) no. 3, pp. 420-432 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled using the well known Iterative Proportional Fitting Procedure. The proposed solution is not an exact mathematical solution of a marginal problem but just its approximation applicable in many practical situations like Monte Carlo sampling. This is why the authors deal not only with the consistent case, when the iterative procedure converges, but also with the inconsistent non-converging case. In the latter situation, the IPF procedure tends to cycle (when combining three pairwise dependencies the procedure creates three convergent subsequences), and thus the authors propose some heuristics yielding a ``solution'' of the problem even for inconsistent pairwise dependence relations.
As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled using the well known Iterative Proportional Fitting Procedure. The proposed solution is not an exact mathematical solution of a marginal problem but just its approximation applicable in many practical situations like Monte Carlo sampling. This is why the authors deal not only with the consistent case, when the iterative procedure converges, but also with the inconsistent non-converging case. In the latter situation, the IPF procedure tends to cycle (when combining three pairwise dependencies the procedure creates three convergent subsequences), and thus the authors propose some heuristics yielding a ``solution'' of the problem even for inconsistent pairwise dependence relations.
Classification : 94A17, 97K50
Keywords: Frank copula; IPFP; entropy 
@article{KYB_2013_49_3_a3,
     author = {B{\'\i}na, Vladislav and Jirou\v{s}ek, Radim},
     title = {A short note on multivariate dependence modeling},
     journal = {Kybernetika},
     pages = {420--432},
     year = {2013},
     volume = {49},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a3/}
}
TY  - JOUR
AU  - Bína, Vladislav
AU  - Jiroušek, Radim
TI  - A short note on multivariate dependence modeling
JO  - Kybernetika
PY  - 2013
SP  - 420
EP  - 432
VL  - 49
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a3/
LA  - en
ID  - KYB_2013_49_3_a3
ER  - 
%0 Journal Article
%A Bína, Vladislav
%A Jiroušek, Radim
%T A short note on multivariate dependence modeling
%J Kybernetika
%D 2013
%P 420-432
%V 49
%N 3
%U http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a3/
%G en
%F KYB_2013_49_3_a3
Bína, Vladislav; Jiroušek, Radim. A short note on multivariate dependence modeling. Kybernetika, Tome 49 (2013) no. 3, pp. 420-432. http://geodesic.mathdoc.fr/item/KYB_2013_49_3_a3/

[1] Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula construction of multiple dependence. Insurance Math. Econom. 44, 2 (2009), 182-198. | DOI | MR

[2] Asci, C., Piccioni, M.: A note on the IPF algorithm when the marginal problem is unsolvable. Kybernetika 39 (2003), 6, 731-737. | MR | Zbl

[3] Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975), 146-158. | DOI | MR | Zbl

[4] Deming, W. E., Stephan, F. F.: On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), 427-444. | DOI | MR

[5] Hennessy, D. A., Lapan, H. E.: The use of Archimedean copulas to model portfolio allocations. Math. Finance 12 (2002), 2, 143-154. | DOI | MR

[6] Jiroušek, R.: Solution of the marginal problem and decomposable distributions. Kybernetika 27, 5 (1991), 403-412. | MR | Zbl

[7] Kratochvíl, V.: Characteristic properties of equivalent structures in compositional models. | Zbl

[8] Li, D. X.: On default correlation: A copula function approach. J. Fixed Income 9 (2000), 4, 43-54. | DOI

[9] Mareš, M., Mesiar, R.: Aggregation of complex quantities. In: Proceedings of AGOP'2005. International Summer School on Aggregation Operators and Their Applications (R. Mesiar, G. Pasi, and M. Faré, eds.), Universitá della Svizzeria Italiana, Lugano 2005, pp. 85-88.

[10] Rüschendorf, L.: Convergence of the iterative proportional fitting procedure. Ann. Statist. 23 (1995), 4, 1160-1174. | DOI | MR | Zbl

[11] Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231 . | MR

[12] Schirmacher, D., Schirmacher, E.: Multivariate Dependence Modeling Using Pair-copulas. Technical Report, Society of Acturaries, Enterprise Risk Management Symposium, Chicago 2008.

[13] Vomlel, J.: Integrating inconsistent data in a probabilistic model. J. Appl. Non-Classical Logics 14 (2004), 3, 367-386. | DOI | Zbl

[14] Weiss, G. N. F.: Copula parameter estimation: numerical considerations and implications for risk management. J. Risk 13 (2010), 1, 17-53.