Exchangeable exogenous shock models

Abstract

We characterize a comprehensive family of d-variate exogenous shock models. Analytically, we consider a family of multivariate distribution functions that arises from ordering, idiosyncratically distorting, and finally multiplying the arguments. Necessary and sufficient conditions on the involved distortions to yield a multivariate distribution function are given. Probabilistically, the attainable set of distribution functions corresponds to a large class of exchangeable exogenous shock models. Besides, the vector of exceedance times of an increasing additive stochastic process across independent exponential trigger variables is shown to constitute an interesting subclass of the considered distributions and yields a second probabilistic model. The alternative construction is illustrated in terms of two examples.

Copulas Based on Marshall–Olkin Machinery

“…Related random samples are depicted in Figure 3. Another example of copulas of type (5) is given by the bivariate Sato copula of [21], generated by F(t) = (2 − t 1/α ) −α for every α > 0. Copulas of type (5) can be interpreted as the exchangeable (i.e., invariant under permutation of their arguments) members of the family proposed in [23, Proposition 3.1].…”

“…To overcome such oversimplification, a convenient generalization has been provided in [21]. Basically, copulas of type (7) have been extended to the form…”

Exogenous shock models: analytical characterization and probabilistic construction

“…Third, and this is the path we pursue, shock distributions other than the exponential law can be considered. This has been already considered for the bivariate case, see [3,12] as well as for the exchangeable d-variate case, see [4,17]. An interesting result, that was derived in [21], is that the class of distributions, which is characterized by a modified lack-of-memory property, where the generic addition is replaced by a reducible and associative binary operator, is a subgroup of GMO distributions with shocks survival functions of the form exp{−λ I H(t)}.…”

A family of transformed copulas with a singular component

“…Durante et al (2007) presented a generalization of the Archimedean family of bivariate copulas with a singular component along the main diagonal. Mai et al (2016) provided a family of the copula functions interrelated with the set of exchangeable exogenous shock models, and the copula functions have singular components along the main diagonal. Xie et al (2017) introduced a new family of the multivariate copula functions defined by two generators, which is a generalization of the multivariate Archimedean copula family with a singular component along the main diagonal.…”

“…Inclusion of many known copula families. The family of TF copulas includes many known bivariate copulas, such as the copula functions presented in Durante et al (2007), Mai et al (2016) and Xie et al (2017). The TF copula family can be regarded as a generalization of the above copula functions.…”