I am struggling with a pretty basic question in MLE of Gaussian Copula.
I have $N$ data points for a Random variable $Y$, each of which is $P$-dimensional. Each of these points is generated from a Gaussian Copula whose CDF is $C$, and the corresponding Gaussian CDF is $\Psi$ and PDF is $\Phi$ (Subscript $j$ denotes the corresponding marginal CDFs and PDFs respectively).
Say, empirical CDF of the mariginals is $F_{1\leq j \leq P}$ . Using $F$, I can generate $U_{j} = F_{j}(Y_{j})$ and $X_{j} = \Psi^{-1}_{j}(U_{j})$
I can try to maximize the likelihood of these points in the Copula space or the Gaussian Space. What is mean is
In the Copula Space, the likelihood is given by:
\begin{align} L & = \sum_{i=1}^{N} \log \left( \frac{ \Phi \left( \textbf{X}; \mu, \Sigma \right)} {\prod_{j=1}^{P} \Phi_{j}(X_{ij}; \mu_j \Sigma_{jj})} \right) \end{align}
In the Gaussian Space, the likelihood is given by: \begin{align} L & = \sum_{i=1}^{N} \log \left( \Phi \left(\textbf{X} ; \mu, \Sigma\right)\right) \end{align}
Practical difficulties aside, theoretically, in which space should we maximize the likelihood?
Thank you.
Edit: In case if someone finds it unclear about the CDFs of the mulitvariates RVs - $\Phi(\textbf{X}) = \Phi(x_1,x_2,\dots, x_p) = Pr(X_{1} \leq x_1, X_{2} \leq x_2, \dots, X_{P} \leq x_P)$