Multivariate imprecise Sklar type theorems

doi:10.1016/j.fss.2020.12.002

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Multivariate imprecise Sklar type theorems

2022-01-01

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Matjaž Omladič *

Nik Stopar

DOI：10.1016/j.fss.2020.12.002

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Abstract

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The omnipotent instrument for modeling multivariate dependence of random variables in standard probability theory has become copulas discovered by A. Sklar in 1959. Only recently Omladic and Stopar prove that in the bivariate case an analogous role is played by exactly the same copulas for random variables coming from finitely additive probability spaces. One of the main results of this paper is that this is true also in the general multivariate case. The extension to n dimensions requires a better understanding of quasi-copulas, the lattice closure of copulas with respect to the pointwise order. We need to develop a new equivalent definition of this notion that should be useful in other applications as well. Another tool we introduce and seems to be new even in the standard probability approach, is multivariate quasi-distributions. We also expand the coherence theory for quasi-copulas and quasi-distributions to the multivariate situation. Finally, our main result is a multivariate Sklar type theorem in the imprecise setting. (c) 2020 Elsevier B.V. All rights reserved.

Keywords:

Copula

Quasi-copula

Imprecise copula

Distribution

Quasi-distribution

Sklar's theorem

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