One important result of this representation is the ability to simulate exactly Archimedean copula random variates, as discussed in Marshall and Olkin (1988) and shown in Algorithm 11.5.

The following results in Table 11.1 from Hofert (2008, table 1) demonstrate examples of popular Archimedean copula models for which closed form distributions of such inverse Laplace transforms of the generator are known.

Table 11.1 Generators and inverse Laplace transforms for several copulas from Archimedean family

Archimedean Family ρ Range Generator ψ (x; ρ) Distribution of images –1 [ψ]
Ali–Mikhail–Haq [0, 1] images
Clayton (0, ) Γ(1/ ...

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