# Inverse distribution of gaussian mixture

In one of the papers I've encountered (Tewari et. al., p. 288 (7)), 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 $$(\Psi_j)^{-1}$$?

• 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.