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BY 4.0 license Open Access Published by De Gruyter Open Access October 18, 2021

Sklar’s theorem, copula products, and ordering results in factor models

  • Jonathan Ansari EMAIL logo and Ludger Rüschendorf
From the journal Dependence Modeling


We consider a completely specified factor model for a risk vector X = (X1, . . ., Xd), where the joint distributions of the components of X with a risk factor Z and the conditional distributions of X given Z are specified. We extend the notion of *-product of d-copulas as introduced for d = 2 and continuous factor distribution in Darsow et al. [6] and Durante et al. [8] to the multivariate and discontinuous case. We give a Sklar-type representation theorem for factor models showing that these *-products determine the copula of a completely specified factor model. We investigate in detail approximation, transformation, and ordering properties of *-products and, based on them, derive general orthant ordering results for completely specified factor models in dependence on their specifications. The paper generalizes previously known ordering results for the worst case partially specified risk factor models to some general classes of positive or negative dependent risk factor models. In particular, it develops some tools to derive sharp worst case dependence bounds in subclasses of completely specified factor models.

MSC 2010: 60E15; 60E05; 28A50


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Received: 2021-01-19
Accepted: 2021-07-16
Published Online: 2021-10-18

© 2021 Jonathan Ansari et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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