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

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• 作者：Bernard De Baets ; Hans De Meyer ; Juan Fern¨¢ndez S¨¢nchez ; Manuel ¨²beda-Flores
• 关键词：Copula ; Linear programming ; Lipschitz condition ; Quasi-copula ; Simplex method ; Trivariate (quasi-)copula
• 刊名：Fuzzy Sets and Systems
• 出版年：2013
• 出版时间：1 October, 2013
• 年：2013
• 卷：228
• 期：Complete
• 页码：3-14
• 全文大小：276 K

文摘

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.