Final solution to the problem of relating a true copula to an imprecise copula. (English) Zbl 1452.62353
Summary: In this paper we solve in the negative the problem proposed in this journal [I. Montes et al., ibid. 278, 48–66 (2014; Zbl 1377.60033)] whether an order interval defined by an imprecise copula contains a copula. Namely, if \(\mathcal{C}\) is a nonempty set of copulas, then \(\underline{\mathcal{C}} = \inf \{ C \}_{C \in \mathcal{C}}\) and \(\overline{\mathcal{C}} = \sup \{ C \}_{C \in \mathcal{C}}\) are quasi-copulas and the pair \(( \underline{\mathcal{C}}, \overline{\mathcal{C}})\) is an imprecise copula according to the definition introduced in the cited paper, following the ideas of \(p\)-boxes. We show that there is an imprecise copula \((A, B)\) in this sense such that there is no copula \(C\) whatsoever satisfying \(A \leqslant C \leqslant B\). So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of M. Dibala et al. [Kybernetika 52, No. 6, 848–865 (2016; Zbl 1389.60005)] where possibly negative volumes of quasi-copulas as defects from being copulas were studied.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
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