# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 3 Distributional Properties
Then,_  
\[\int_{0}^{\infty}\phi_{p}(\mathbf{x};\mathbf{\xi},u^{-1}\mathbf{\Omega})\Phi( \sqrt{u}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{x}-\mathbf{\xi}))h(u;\
u)\,\mathrm{d}u =t_{p}(\mathbf{x};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_ {\mathbf{\Omega}}(\mathbf{x},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}( \mathbf{x}-\mathbf{\xi});\
u+p\Big{)}. \tag{12}\]  
Proof. : The result follows from the representation (4). **Lemma 2**. : _If \(\mathbf{G}(p\times p)>0\), \(\mathbf{b},\mathbf{c}\in\mathbb{R}^{p}\), and \(k\) is a scalar, then_  
\[(2\pi)^{-p/2}\det(\mathbf{G})^{-1/2}\int_{\mathbb{R}^{p}}\exp\Big{(} -\frac{\mathbf{x}^{\prime}\mathbf{G}^{-1}\mathbf{x}-2\mathbf{b}^{\prime}\mathbf{x}}{2}\Big{)}\Phi (k+\mathbf{c}^{\prime}\mathbf{x})\,\mathrm{d}\mathbf{x}=\exp\Big{(}\frac{\mathbf{b}^{\prime} \mathbf{G}\mathbf{b}}{2}\Big{)}\Phi\Big{(}\frac{k+\mathbf{c}^{\prime}\mathbf{G}\mathbf{b}}{\sqrt{1 +\mathbf{c}^{\prime}\mathbf{G}\mathbf{c}}}\Big{)}.\]  
Proof. : See Azzalini [27] Lemma 5.3. **Theorem 2**. : _Let \(\mathbf{Y}\sim\mathrm{TST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;R)\). Consider the partitions given in (9) and (10). If \(R_{2}=\mathbb{R}^{p_{2}}\), then \(\mathbf{Y}_{1}\sim\mathrm{TST}_{p_{1}}(\mathbf{\xi}_{1},\mathbf{\Omega}_{11},\mathbf{\alpha}_{ 1(2)},\
u;R_{1})\), with \(\mathbf{\alpha}_{1(2)}=(1+\mathbf{a}_{2}^{\prime}\mathbf{\omega}_{2}^{-1}\mathbf{\Omega}_{22 :1}\mathbf{\omega}_{2}^{-1}\mathbf{\alpha}_{2})^{-1/2}(\mathbf{\alpha}_{1}+\mathbf{\omega}_{1} \mathbf{\Omega}_{11}^{-1}\mathbf{\Omega}_{12}\mathbf{\omega}_{2}^{-1}\mathbf{\alpha}_{2})\)._  
Proof. : From (11) and (12), we have  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p_{2}}}\int_{ 0}^{\infty}\phi_{p}(\mathbf{y};\mathbf{\xi},u^{-1}\mathbf{\Omega})\Phi(\sqrt{u}\mathbf{\alpha }^{\prime}\omega^{-1}(\mathbf{y}-\mathbf{\xi}))h(u;\
u)\,\mathrm{d}u\,\mathrm{d}\mathbf{y }_{2}}{\int_{\mathbb{R}}\int_{0}^{\infty}\phi_{p}(\mathbf{y};\mathbf{\xi},u^{-1}\mathbf{ \Omega})\Phi(\sqrt{u}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}-\mathbf{\xi}))h(u;\
u )\,\mathrm{d}u\,\mathrm{d}\mathbf{y}},\quad\mathbf{y}_{1}\in R_{1}.\]  
By noting \(\phi_{p}(\mathbf{y};\mathbf{\xi},u^{-1}\mathbf{\Omega})=\phi_{p_{1}}(\mathbf{y}_{1};\mathbf{\xi}_{1 },u^{-1}\mathbf{\Omega}_{11})\phi_{p_{2}}(\mathbf{y}_{2};\mathbf{\xi}_{2\cdot 1},u^{-1}\mathbf{\Omega}_{22:1})\) and by applying [PERSON]'s theorem, we have  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{0}^{\infty}\phi_{p_{1}}( \mathbf{y}_{1};\mathbf{\xi}_{1},u^{-1}\mathbf{\Omega}_{11})\Big{[}\int_{\mathbb{R}^{p_{2}} }\phi_{p_{2}}(\mathbf{y}_{2};\mathbf{\xi}_{2\cdot 1},u^{-1}\mathbf{\Omega}_{22:1})\Phi(\sqrt{u}\mathbf{\alpha}^{ \prime}\omega^{-1}(\mathbf{y}-\mathbf{\xi}))\,\mathrm{d}\mathbf{y}_{2}\Big{]}h(u;\
u)\, \mathrm{d}u}{\int_{R_{1}}\int_{0}^{\infty}\phi_{p_{1}}(\mathbf{y}_{1};\mathbf{\xi}_{1 },u^{-1}\mathbf{\Omega}_{11})\Big{[}\int_{\mathbb{R}^{p_{2}}}\phi_{p_{2}}(\mathbf{y}_{2}; \mathbf{\xi}_{2\cdot 1},u^{-1}\mathbf{\Omega}_{22:1})\Phi(\sqrt{u}\mathbf{\alpha}^{\prime}\omega^{ -1}(\mathbf{y}-\mathbf{\xi}))\,\mathrm{d}\mathbf{y}_{2}\Big{]}h(u;\
u)\,\mathrm{d}u\, \mathrm{d}\mathbf{y}_{1}}. \tag{13}\]  
To evaluate the integral with respect to \(\mathbf{y}_{2}\), we use the identity \(\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}-\mathbf{\xi})=(1+\mathbf{a}_{2}^{\prime}\mathbf{ \omega}_{2}^{-1}\mathbf{\Omega}_{22:1}\mathbf{\omega}_{2}^{-1}\mathbf{\alpha}_{2})^{1/2}\mathbf{ \alpha}_{1(2)}^{\prime}\mathbf{\omega}_{1}^{-1}(\mathbf{y}_{1}-\mathbf{\xi}_{1})+\mathbf{\alpha}_{ 2}^{\prime}\mathbf{\omega}_{2}^{-1}(\mathbf{y}_{2}-\mathbf{\xi}_{2\cdot 1})\) and subsequently change the variable \(\mathbf{x}=\mathbf{y}_{2}-\mathbf{\xi}_{2\cdot 1}\).