Non-exchangeability of copulas arising from shock models

Abstract Copula is a useful tool that captures the dependence structure among random variables. In practice, it is an important question which copula to choose depending on the given data and stochastic assumptions on the model in order to achieve an appropriate interpretation of the data at hand. This paper intends to help a practitioner to make a better decision about that. We concentrate on the study of the lack of exchangeability, a copulas’ attribute closely studied only recently. The main non-exchangeability measure μ ∞ for a family of copulas is the supremum of the differences | C ( x , y ) − C ( y , x ) | over all ( x , y ) and all copulas C in the family. We give the sharp bound of μ ∞ for the families of Marshall copulas, maxmin and reflected maxmin copulas (i.e. the main shock-model based copulas) as well as the families of positively and of negatively quadrant dependent copulas. A major contribution of this paper is also exact calculation of the maximal asymmetry function on each of the particular families of copulas. When restricted to special families of copulas considered, it helps us finding the sharp bound of μ ∞ for each of the given families. And even more importantly, it helps us giving a stochastic interpretation of the extremal copulas and examples of shock models where the maximal asymmetry is attained.