Reparameterizing Marshall–Olkin copulas with applications to sampling

It is shown that exchangeable Marshall–Olkin survival copulas coincide with a parametric family of copulas studied in [J.-F. Mai and M. Scherer, Lévy-Frailty copulas, J. Multivariate Anal. 100 (2009), pp. 1567–1585]. This observation implies an alternative probabilistic interpretation in many cases and allows the transfer of known results from one family to the other. For instance, using the classical construction of [A.W. Marshall and I. Olkin, A multivariate exponential distribution, J. Am. Stat. Assoc. 62 (1967), pp. 30–44], sampling an n-dimensional Marshall–Olkin copula requires 2 n −1 exponentially distributed random variables, which is inefficient in large dimensions. Applying the alternative construction, sampling an exchangeable n-dimensional copula boils down to generating n independent exponentially distributed random variables and one path of a certain Lévy subordinator, which is highly efficient in many cases. Furthermore, the alternative model and sampling methodology is generalized to high-dimensional hierarchical copulas. A sampling algorithm for the latter is described in detail and illustrated with an example.