Constructing copulas from shock models with imprecise distributions. (English) Zbl 1471.62360
Summary: The omnipotence of copulas when modeling dependence given marginal distributions in a multivariate stochastic situation is assured by the Sklar’s theorem. I. Montes et al. [Fuzzy Sets Syst. 278, 48–66 (2015; Zbl 1377.60033)] suggest the notion of what they call an 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 [the first author and N. Stopar, Fuzzy Sets Syst. 393, 96–112 (2020; Zbl 1452.62353)] 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.
MSC:
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
| 62C20 | Minimax procedures in statistical decision theory |
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