# Final solution to the problem of relating a true copula to an imprecise copula

## TL;DR

This paper demonstrates that the existing definition of imprecise copulas does not always contain a true copula within the specified bounds, challenging prior assumptions and offering new insights into quasi-copulas.

## Contribution

It provides a counterexample showing the limitations of the current imprecise copula definition and introduces methods to analyze quasi-copulas' defects.

## Key findings

- Existence of imprecise copulas with no containing copula
- Counterexample challenges previous definitions
- Methods to analyze quasi-copula defects

## Abstract

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 $\mathcal{C}$ is a nonempty set of copulas, then $\underline{C} = \inf\{C\}_{C\in\mathcal{C}}$ and $\overline{C}= \sup\{C\}_{C\in\mathcal{C}}$ are quasi-copulas and the pair $(\underline{C},\overline{C})$ 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 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.