Suppose the distribution of an $n$-dimensional random vector $X$ is characterized by a Gaussian copula $C_R$ with correlation matrix $R$ and a set of marginal $\{(F_{X_i}, f_{X_i})\}_{i=1}^n$ (pairs of a distribution function and a density function). I would like to derive the probability density of $X$. Wikipedia says that the density of the copula is as follows:
$$
f_{C_R}(u)
= \frac{1}{\sqrt{\det{R}}}\exp\left(-\frac{1}{2}
\begin{pmatrix}\Phi^{-1}(u_1)\\ \vdots \\ \Phi^{-1}(u_n)\end{pmatrix}^T \cdot
\left(R^{-1}-I\right) \cdot
\begin{pmatrix}\Phi^{-1}(u_1)\\ \vdots \\ \Phi^{-1}(u_n)\end{pmatrix}
\right)
$$
Given this information, I decided to apply a change-of-variables transformation and, thereby, express the density of $X$ in terms of the density of $U \sim C_R$. I arrived at the following formula:
$$
f_X(x) = \left| \prod_{i=1}^n f_{X_i}(x_i) \right| \, \, f_{C_R}\left(
\begin{pmatrix}F_{X_1}(x_1)\\ \vdots \\ F_{X_n}(x_n)\end{pmatrix}
\right)
$$
The coefficient before $f_{C_R}$ is the absolute value of the determinant of the Jacobian of the inverse of the transformation from $U$ to $X$, which is just an application of $\{F^{-1}_{X_i}\}$ to the corresponding elements of $U$.
Since I do not have a proper background, I am not entirely sure should that all the steps that I have taken are legitimate. I, therefore, would be grateful if somebody could confirm that the above computations are correct.
I also wonder why $f_{C_R}$ differs so much from the density of a centered multivariate Gaussian vector:
$$
f_Y(y) =
\frac{1}{\sqrt{(2\pi)^n \det{\Sigma}}}
\exp\left(-\frac{1}{2}y^\mathrm{T}\Sigma^{-1}y
\right).
$$
How did $-I$ in $f_{C_R}$ come about? And why does not $f_{C_R}$ have $\sqrt{(2\pi)^n}$? It would be nice if somebody could explain this.
Thank you!
Regards,
Ivan
Best Answer
The Gaussian copula (with correlation matrix $R$): $$ C_R(u)=\Phi_n(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_n);R). $$
Then, letting $\zeta:=(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_n))^{\top}$, $$ c_R(u)=\frac{\phi_n(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_n);R)}{\phi(\Phi^{-1}(u_1))\cdots \phi(\Phi^{-1}(u_1))} \\=(2\pi)^{-n/2}\lvert R \rvert^{-1/2}\frac{-\exp(\zeta^{\top}R^{-1}\zeta/2)}{\prod_{i=1}^n (2\pi)^{-1/2}\exp(-\zeta_i^2/2)}=\lvert R \rvert^{-1/2}\frac{\exp(-\zeta^{\top}R^{-1}\zeta/2)}{\exp(-\zeta^{\top}I_n\zeta/2)}. $$
Here, $\Phi_n(\cdot;R),\phi_n(\cdot;R)$ are zero-mean multivariate normal CDF and PDF with covariance matrix $R:n\times n$.