# Constructing copulas from shock models with imprecise distributions

## TL;DR

This paper extends the concept of imprecise copulas to shock models with uncertain marginals, introducing stronger conditions and a new stochastic order to better model dependence under imprecision.

## Contribution

It introduces imprecise copulas derived from shock models with imprecise marginals, establishing stronger conditions and a novel stochastic order for these bivariate functions.

## Key findings

- Imprecise copulas satisfy a stronger condition than standard imprecise copulas.
- A new stochastic order on bivariate objects is developed.
- The approach enhances modeling dependence with uncertain marginals.

## Abstract

The omnipotence of copulas when modeling dependence given marg\-inal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al.\ (2015) suggest the notion of what they call an \emph{imprecise copula} that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of $p$-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladi\v{c} and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects.