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

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 co...

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Autores principales:	Ansari Jonathan, Rüschendorf Ludger
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2021
Materias:	componentwise convex copulas concordance order conditional distribution function conditional independence factor model product of copulas 60e15 60e05 28a50 Science (General) Q1-390 Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3284530526974815a0e3b0b7e9593405
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spelling	oai:doaj.org-article:3284530526974815a0e3b0b7e95934052021-12-05T14:10:46ZSklar’s theorem, copula products, and ordering results in factor models2300-229810.1515/demo-2021-0113https://doaj.org/article/3284530526974815a0e3b0b7e95934052021-10-01T00:00:00Zhttps://doi.org/10.1515/demo-2021-0113https://doaj.org/toc/2300-2298We 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.Ansari JonathanRüschendorf LudgerDe Gruyterarticlecomponentwise convex copulasconcordance orderconditional distribution functionconditional independencefactor modelproduct of copulas60e1560e0528a50Science (General)Q1-390MathematicsQA1-939ENDependence Modeling, Vol 9, Iss 1, Pp 267-306 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	componentwise convex copulas concordance order conditional distribution function conditional independence factor model product of copulas 60e15 60e05 28a50 Science (General) Q1-390 Mathematics QA1-939
spellingShingle	componentwise convex copulas concordance order conditional distribution function conditional independence factor model product of copulas 60e15 60e05 28a50 Science (General) Q1-390 Mathematics QA1-939 Ansari Jonathan Rüschendorf Ludger Sklar’s theorem, copula products, and ordering results in factor models
description	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.
format	article
author	Ansari Jonathan Rüschendorf Ludger
author_facet	Ansari Jonathan Rüschendorf Ludger
author_sort	Ansari Jonathan
title	Sklar’s theorem, copula products, and ordering results in factor models
title_short	Sklar’s theorem, copula products, and ordering results in factor models
title_full	Sklar’s theorem, copula products, and ordering results in factor models
title_fullStr	Sklar’s theorem, copula products, and ordering results in factor models
title_full_unstemmed	Sklar’s theorem, copula products, and ordering results in factor models
title_sort	sklar’s theorem, copula products, and ordering results in factor models
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/3284530526974815a0e3b0b7e9593405
work_keys_str_mv	AT ansarijonathan sklarstheoremcopulaproductsandorderingresultsinfactormodels AT ruschendorfludger sklarstheoremcopulaproductsandorderingresultsinfactormodels
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Sklar’s theorem, copula products, and ordering results in factor models

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