gn*}
\Pr(\boldsymbol{X} \le \boldsymbol{x}) = \exp \left[ -\sum_{j=1}^d
\frac{1}{x_j} \Phi_{d-1, \boldsymbol{\Sigma}_{-j}} \left( \lambda_{ij}-
\frac{1}{2\lambda_{ij}}  \log\left(\frac{x_j}{x_i}\right), i \in
I_{-j}\right)\right].
\end{align*}\]</span> where the partial correlation matrix <span class="math inline">\(\boldsymbol{\Sigma}_{-j}\)</span> has elements
<span class="math display">\[\begin{align*}
\varrho_{i,k; j}=
\frac{\lambda_{ij}^2+\lambda_{kj}^2-\lambda_{ik}^2}{2\lambda_{ij}\lambda_{kj}}
\end{a