Say I have a (generally high-dimensional) random variable $X$ with known, continuous CDF $F(X)$.
Is there a good algorithm for drawing values of $X$ that doesn't require that I calculate the joint density?
I'm specifically interested in the GEV distribution used in the latent variable formulation of the nested logit: $$F\left(\vec\nu\right) = \exp\left(- ∑_{n ∈ N} \left(∑_{k∈n} \exp\left(\frac{-\nu_k}{λ_n}\right)\right)^{λ_n}\right).$$ That distribution should have analytic PDFs, but they're pretty gross (and are going to get even grosser with multiple layers of nests).
[Edit: this question was labelled a duplicate of Inverse of cumulative density function for Multivariate Normal Distribution. It's not obvious why -- is the assumption that the inverse-transform algorithm is the only algorithm for drawing from a CDF? That doesn't seem right.]