In this paper we solve in the negative the problem proposed in this journal
(I. Montes et al., Sklar's theorem in an imprecise setting, Fuzzy Sets and
Systems, 278 (2015), 48-66) whether an order interval defined by an imprecise
copula contains a copula. Namely, if C is a nonempty set of
copulas, then C=inf{C}C∈C and C=sup{C}C∈C are quasi-copulas and the pair
(C,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⩽C⩽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 Dibala et al. (Defects and
transformations of quasi-copulas, Kybernetika, 52 (2016), 848-865) where
possibly negative volumes of quasi-copulas as defects from being copulas were
studied.Comment: 20 pages; added Conclusion, added some clarifications in proofs,
added some explanations at the beginning of each section, corrected typos,
results remain the sam