# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 3 Distributional Properties
The random vector \(\mathbf{Y}\in A\) has a truncated multivariate extended skew-\(t\) distribution with support \(A\) and parameters \(\mathbf{\xi}\in\mathbb{R}^{p}\), \(\mathbf{\Omega}(p\times p)>0\), \(\mathbf{\alpha}\in\mathbb{R}^{p}\), extension parameter \(\tau\in\mathbb{R}\) and \(\
u>0\) degrees of freedom, denoted by \(\mathbf{Y}\sim\mathrm{TEST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u;A)\), if its PDF is  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p}_{2}}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}_{2}}{\int_{R}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}}, \tag{11}\]1  
where \(\mathbf{y}\in A\). The PDF of \(\mathbf{Y}\sim\mathrm{TEST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u;A)\) can be expressed as  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p}_{2}}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}_{2}}{\int_{R}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}}, \tag{11}\]2  
where \(\mathrm{EST}_{p}(A;\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u)=\int_{A}\mathrm{ EST}_{p}(\mathbf{y};\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u)\,\mathrm{d}\mathbf{y}\), with \(\mathrm{EST}_{p}(\mathbf{x};\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u)\) being the PDF of \(\mathbf{X}\), \(\mathbf{X}\sim\mathrm{EST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u)\) (Definition 2). For \(A=\mathbb{R}^{p}\) in (15), we have the PDF of a multivariate extended skew-\(t\) distribution given in (6). If \(\tau=0\) in (14), we obtain the PDF of a truncated multivariate skew-\(t\) distribution given in (7). The truncated multivariate \(t\) distribution is a limiting case of the truncated multivariate extended skew-\(t\) distribution when \(\tau\to\infty\). The truncated multivariate extended skew-normal distribution ([PERSON] et al. [9]) is obtained from (14) when \(v\to\infty\). If we take \(\mathbf{\alpha}=\mathbf{0}\) in (14), we obtain a new truncated elliptical distribution with the PDF  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p}_{2}}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}_{2}}{\int_{R}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}}, \tag{11}\]3  
which is also an extension of the distribution presented in equation (2) of Arellano-Valle and Genton [3]. Furthermore, the PDF of the truncated multivariate \(t\) distribution can also be obtained from (14) when \(\mathbf{\alpha}=\mathbf{0}\) and \(\tau=0\). In Theorem 3, we derive the conditional distribution of \(\mathbf{Y}_{2}|\mathbf{Y}_{1}\), when \(\mathbf{Y}=(\mathbf{Y}_{1}^{\prime},\mathbf{Y}_{2}^{\prime})^{\prime}\) has a truncated multivariate skew-\(t\) distribution. **Theorem 3**. : _Let \(\mathbf{Y}\sim\mathrm{TS}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;R)\). Consider the partitions given in (9) and (10).