Academic Paper attributes
When choosing the right copula for our data a key point is to distinguish the family that describes it at the best. In this respect, a better choice of the copulas could be obtained through the information about the (non)symmetry of the data. Exchangeability as a probability concept (first next to independence) has been studied since 1930s, copulas have been studied since 1950s, and even the most important class of copulas from the point of view of applications, i.e. the ones arising from shock models s.a. Marshalls copulas, have been studied since 1960s. However, the point of non-exchangeability of copulas was brought up only in 2006 and has been intensively studied ever since. One of the main contributions of this paper is the maximal asymmetry function for a family of copulas. We compute this function for the major families of shock-based copulas, i.e. Marshall, maxmin and reflected maxmin (RMM for short) copulas and also for some other important families. We compute the sharp bound of asymmetry measure muinfty, the most important of the asymmetry measures, for the family of Marshall copulas and the family of maxmin copulas, which both equal to frac427 (approx 0.148). One should compare this bound to the one for the class of PQD copulas to which they belong, which is 3-2sqrt2 approx 0.172), and to the general bound for all copulas that is frac13. Furthermore, we give the sharp bound of the same asymmetry measure for RMM copulas which is 3-2sqrt2, compared to the same bound for NQD copulas, where they belong, which is sqrt5-2 (approx 0.236). One of our main results is also the statistical interpretation of shocks in a given model at which the maximal asymmetry measure bound is attained. These interpretations for the three families studied are illustrated by examples that should be helpful to practitioners when choosing the model for their data.
