# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 3 Distributional Properties
The result stated in the above corollary can also be obtained as a particular case of Theorem 3.3 of [PERSON] and Ferrari [8]. Next, we present results on marginal and conditional distributions involving sub-vectors of the random vector having truncated multivariate skew-\(t\) distribution. For this, we consider partitions of \(\mathbf{Y}\in\mathbb{R}^{p}\), \(\mathbf{\xi}\in\mathbb{R}^{p}\), \(\mathbf{\alpha}\in\mathbb{R}^{p}\), and \(\mathbf{\Omega}(p\times p)>0\) as follows  
\[\mathbf{Y}=(\mathbf{Y}_{1}^{\prime},\mathbf{Y}_{2}^{\prime})^{\prime},\quad \mathbf{\xi}=(\mathbf{\xi}_{1}^{\prime},\mathbf{\xi}_{2}^{\prime})^{\prime},\quad\mathbf{\Omega }= \begin{bmatrix}\mathbf{\Omega}_{11}&\mathbf{\Omega}_{12}\\ \mathbf{\Omega}_{21}&\mathbf{\Omega}_{22}\end{bmatrix} ,\quad\mathbf{\alpha}=(\mathbf{\alpha}_{1 }^{\prime},\mathbf{\alpha}_{2}^{\prime})^{\prime}, \tag{9}\]  
where \(\mathbf{Y}_{1}\in\mathbb{R}^{p_{1}}\), \(\mathbf{Y}_{2}\in\mathbb{R}^{p_{2}}\), \(\mathbf{\xi}_{1}\in\mathbb{R}^{p_{1}}\), \(\mathbf{\xi}_{2}\in\mathbb{R}^{p_{2}}\), \(\mathbf{\alpha}_{1}\in\mathbb{R}^{p_{1}}\), \(\mathbf{\alpha}_{2}\in\mathbb{R}^{p_{2}}\), \(\mathbf{\Omega}_{11}(p_{1}\times p_{1})>0\), \(\mathbf{\Omega}_{22}(p_{2}\times p_{2})>0\), and \(\mathbf{\Omega}_{12}(p_{1}\times p_{2})\) and \(\mathbf{\Omega}_{21}(p_{2}\times p_{1})\) are such that \(\mathbf{\Omega}_{12}^{\prime}=\mathbf{\Omega}_{21}\). In addition, we define \(\mathbf{\omega}_{1}=(\mathbf{\Omega}_{11}\odot I_{p_{1}})^{1/2}\) and \(\mathbf{\omega}_{2}=(\mathbf{\Omega}_{22}\odot I_{p_{2}})^{1/2}\). The Schur complement of the block \(\mathbf{\Omega}_{11}\) of the matrix \(\mathbf{\Omega}\) is given by \(\mathbf{\Omega}_{2:1}=\mathbf{\Omega}_{22}-\mathbf{\Omega}_{21}\mathbf{\Omega}_{11}^{-1}\mathbf{ \Omega}_{12}\). In addition, we define \(\mathbf{\xi}_{2:1}=\mathbf{\xi}_{2}+\mathbf{\Omega}_{21}\mathbf{\Omega}_{11}^{-1}(\mathbf{y}_{1}- \mathbf{\xi}_{1})\). The dimension \(p\) is such that \(p=p_{1}+p_{2}\). We assume that the support set of the truncated multivariate skew-\(t\) distribution is a rectangle \(R\subseteq\mathbb{R}^{p}\), which can be written as \(R=I_{1}\times\cdots\times I_{p}\), where \(I_{1},\ldots,I_{p}\) are finite or infinite intervals. Furthermore, \(R\) can be expressed as  
\[R=R_{1}\times R_{2}, \tag{10}\]  
where \(R_{1}=I_{1}\times\cdots\times I_{p_{1}}\subseteq\mathbb{R}^{p_{1}}\) and \(R_{2}=I_{p_{1}+1}\times\cdots\times I_{p}\subseteq\mathbb{R}^{p_{2}}\). Now, consider the partitions given in (9) and (10) for \(\mathbf{Y}\sim\mathrm{TST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;R)\). Integrating (7) with respect to \(\mathbf{y}_{2}\), we find the marginal PDF of \(\mathbf{Y}_{1}\) as  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p}_{2}}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}_{2}}{\int_{R}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}}, \tag{11}\]  
where \(\mathbf{y}_{1}\in R_{1}\). It is noteworthy that the above PDF does not necessarily have the structure of the PDF (7). In Theorem 2, we establish conditions on the support set \(R\) for some marginals to preserve the truncated multivariate skew-\(t\) distribution. To proceed, we need the following preliminary results. **Lemma 1**. : _Let \(h(u;\
u)\) be the PDF given in (5).