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In one of the papers I've encountered, the authors propose a copula function

$$ c(u_1, \ldots, u_d; \Theta) = \frac{\psi(y_1, \ldots, y_d; \Theta)}{\prod_{j=1}^{d}\psi_j (y_j)}$$

where $\psi(y_1, \ldots, y_d; \Theta) = \sum_k \pi_k \phi(x_1, \ldots, x_d; \theta_k)$ is a Gaussian Mixture Density with certain weights $\pi_j$; $\psi_j$ are it's appropriate marginals, and $y_j= \Psi^{-1}_j(u_j)$, where $\Psi_j^{-1}$ is the inverse distribution function of the Gaussian Mixture along the $j-$th dimension.

However, it's not clear how do we, given a certain mixture function, would express the inverse of $\Psi_j^{-1}$.

Following this answer it would seem that in general the quantile function of Gaussian Mixture does not have an analytical form, but what about its $j-$th marginal (if I understand this paper correctly)? Do we know its expression in general?

How should we understand "along the j-th dimension"? Is it $(\Psi^{-1})_j$ or $(\Psi_j)^{-1}$?

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  • $\begingroup$ @Xi'an the initial intention behind the link was to give a source where the question is coming from, but I will remove it if it seems as a requirement. $\endgroup$ – Nutle May 23 at 11:48
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It is the marginal of the mixture of normals, by definition of copula function:

https://en.wikipedia.org/wiki/Copula_(probability_theory)

This is, the quantile function corresponding to a marginal mixture of normals, where the marginal corresponds to the entry of interest.

Unless you were reading Sklar's paper, this corresponds to the definition of copula density function rather than a new model. The new part is the use of Gaussian mixtures.

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  • $\begingroup$ Thanks, but this does not answer any of the questions (regarding the analytical form of Gaussian Mixture quantiles and $(\Psi^{-1})_j$ vs $(\Psi_j)^{-1}$). The cited paper can be found in the "Edits" of the question - removed it following a @Xi'an's remark. $\endgroup$ – Nutle Jun 14 at 9:32

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