How to simulate from a log-copula function?

Question

Does anybody know how to simulate from a log-copula function?

I'm trying to simulate $(u,v)$ from a log-copula function with the CDF:
$$ C(u,v, a) = \exp\bigg(1-\big[(1 - \ln u)^a + (1 - \ln v)^a - 1\big]^{1/a}\bigg) $$ I guess I should use the Marshal-Olkin method, but don't know how.

Answer 1

Example 1

Simulating from the copula $$C_0(u,v, \theta) = (u^{-θ} + v^{-θ} − 1)^{-1/θ}$$ can be done by the following algorithm (see also Embrechts et al., 2001 for much more fascinating details):

1. Generate $U,V\stackrel{\text{iid}}{\sim}\mathcal{U}(0,1)$
2. Take $W=(\{V^{-\theta/(1+\theta)}-1\}U^{-\theta}+1)^{-1/\theta}$
3. Return $(U,W)$

Now your copula is the transform $$C_1(u,v,a)=\exp\{1-C_0(1-\log u,1-\log v,-a)\}\,.$$

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For an Archimedian copula of the form $$C(u,v)=\varphi^{-1}(\varphi(u)+\varphi(v))$$it can be shown that the pair$$S=\varphi(U)/\{\varphi(U)+\varphi(V)\},\ T=C(U,V)$$is associated with the joint cdf$$G(s,t)=s\{t-\varphi(t)/\varphi^\prime(t)\}=sK(t)$$(see Wu et al., 2006) which means $S$ and $T$ are independent and can be simulated as

1. Generate $R,S\stackrel{\text{iid}}{\sim}\mathcal{U}(0,1)$
2. Take $T=K^{-1}(R)$
3. Take $U=\varphi^{-1}(S\varphi(T))$ and $V=\varphi^{-1}((1-S)\varphi(T))$