On the existence of a trivariate copula with given values of a trivariate quasi-copula at several points

Authors:
Bernard De Baets;Hans De Meyer;Juan FernáNdez SáNchez;Manuel íBeda-Flores
Affiliations:
Department of Mathematical Modelling, Statistics and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent, Belgium;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 S9, B-9000 Gent, Belgium;Research Group of Mathematical Analysis, University of Almería, Carretera de Sacramento s/n, 04120 Almería, Spain;Department of Statistics and Applied Mathematics, University of Almería, Carretera de Sacramento s/n, 04120 Almería, Spain
Venue:
Fuzzy Sets and Systems
Year:
2013

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Abstract

The existence of a (multivariate) copula with a given value of a (multivariate) quasi-copula at a single point is known. In the bivariate case, the existence of a copula with given values of a quasi-copula at two or three arbitrary points is also known. In this paper, we give an alternative proof of the existence of a trivariate copula with a given value of a trivariate quasi-copula at a single point. This proof relies on the reformulation of the existence problem as a linear programming minimization problem and its solution by means of the simplex algorithm. The same method is then used to prove the existence of a trivariate copula with given values of a trivariate quasi-copula at two arbitrary points in the unit cube. We furthermore establish a counter-example showing that the existence for given values at three points is not guaranteed. This completes the analysis of the trivariate case.