Title: | A short note on multivariate dependence modeling (English) |

Author: | Bína, Vladislav |

Author: | Jiroušek, Radim |

Language: | English |

Journal: | Kybernetika |

ISSN: | 0023-5954 |

Volume: | 49 |

Issue: | 3 |

Year: | 2013 |

Pages: | 420-432 |

Summary lang: | English |

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Category: | math |

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Summary: | 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. (English) |

Keyword: | Frank copula |

Keyword: | IPFP |

Keyword: | entropy |

MSC: | 94A17 |

MSC: | 97K50 |

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Date available: | 2013-07-18T15:32:54Z |

Last updated: | 2013-07-31 |

Stable URL: | http://hdl.handle.net/10338.dmlcz/143356 |

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Reference: | [1] Aas, K., Czado, C., Frigessi, A., Bakken, H.: Pair-copula construction of multiple dependence..Insurance Math. Econom. 44, 2 (2009), 182-198. MR 2517884, 10.1016/j.insmatheco.2007.02.001 |

Reference: | [2] Asci, C., Piccioni, M.: A note on the IPF algorithm when the marginal problem is unsolvable..Kybernetika 39 (2003), 6, 731-737. Zbl 1245.62070, MR 2035647 |

Reference: | [3] Csiszár, I.: I-divergence geometry of probability distributions and minimization problems..Ann. Probab. 3 (1975), 146-158. Zbl 0318.60013, MR 0365798, 10.1214/aop/1176996454 |

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Reference: | [6] Jiroušek, R.: Solution of the marginal problem and decomposable distributions..Kybernetika 27, 5 (1991), 403-412. Zbl 0752.60009, MR 1132602 |

Reference: | [7] Kratochvíl, V.: Characteristic properties of equivalent structures in compositional models.. Zbl 1214.68400 |

Reference: | [8] Li, D. X.: On default correlation: A copula function approach..J. Fixed Income 9 (2000), 4, 43-54. 10.3905/jfi.2000.319253 |

Reference: | [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. |

Reference: | [10] Rüschendorf, L.: Convergence of the iterative proportional fitting procedure..Ann. Statist. 23 (1995), 4, 1160-1174. Zbl 0851.62038, MR 1353500, 10.1214/aos/1176324703 |

Reference: | [11] Sklar, A.: Fonctions de répartition à n dimensions et leurs marges..Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231 . MR 0125600 |

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

Reference: | [13] Vomlel, J.: Integrating inconsistent data in a probabilistic model..J. Appl. Non-Classical Logics 14 (2004), 3, 367-386. Zbl 1185.68699, 10.3166/jancl.14.367-386 |

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

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