# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 3 Distributional Properties  
In Theorem 1, we state and prove the closure property of the truncated multivariate skew-\(t\) distribution under affine transformations. **Theorem 1**. : _Let \(T:\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}\) be the transformation \(T(\mathbf{x})=\mathbf{\lambda}+\mathbf{\Delta x}\), being \(\mathbf{\lambda}\in\mathbb{R}^{p}\) a constant vector and \(\mathbf{\Delta}(p\times p)\) a non-singular constant matrix. If \(\mathbf{Y}\sim\mathrm{TST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;A)\), then \(T(\mathbf{Y})=\mathbf{\lambda}+\mathbf{\Delta Y}\sim\mathrm{TST}_{p}(\widetilde{\mathbf{\xi}}, \widetilde{\mathbf{\Omega}},\widetilde{\mathbf{\omega}},\
u;T(A))\), where \(\widetilde{\mathbf{\xi}}=\mathbf{\lambda}+\mathbf{\Delta\xi}\), \(\widetilde{\mathbf{\Omega}}=\mathbf{\Delta\Omega\Delta^{\prime}}\), and \(\widetilde{\mathbf{\alpha}}=\widetilde{\mathbf{\omega}}(\mathbf{\Delta}^{-1})^{\prime}\mathbf{ \omega}^{-1}\mathbf{\alpha}\), with \(\widetilde{\mathbf{\omega}}=(\widetilde{\mathbf{\Omega}}\odot I_{p})^{1/2}\)._  
Proof. : Applying the transformation \(\mathbf{W}=T(\mathbf{Y})=\mathbf{\lambda}+\mathbf{\Delta Y}\) in (7) with the Jacobian \(J(\mathbf{y}\rightarrow\mathbf{w})=\det(\mathbf{\Delta})^{-1}\), we have  
\[f_{\mathbf{W}}(\mathbf{w})=\frac{t_{p}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{ \lambda});\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{ \mathbf{\Omega}}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda}),\mathbf{\xi})}\Big{]}^{1/2}\mathbf{ \alpha}^{\prime}\mathbf{\omega}^{-1}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda})-\mathbf{ \xi});\
u+p\Big{)}}{\int_{T(A)}t_{p}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{ \lambda});\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{ \mathbf{\Omega}}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda}),\mathbf{\xi})}\Big{]}^{1/2}\mathbf{ \alpha}^{\prime}\mathbf{\omega}^{-1}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda})-\mathbf{ \xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{w}},\]  
where \(\mathbf{w}\in T(A)\). By noting \(\delta_{\mathbf{\Omega}}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda}),\mathbf{\xi})=\delta_{ \widetilde{\mathbf{\Omega}}}(\mathbf{w},\widetilde{\mathbf{\xi}})\), \(t_{p}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda});\mathbf{\xi},\mathbf{\Omega},\
u)=\det( \mathbf{\Delta})t_{p}(\mathbf{w};\widetilde{\mathbf{\xi}},\widetilde{\mathbf{\Omega}},\
u)\) and \(\mathbf{\alpha}^{\prime}\mathbf{\omega}^{-1}(\mathbf{\Delta}^{-1}(\mathbf{w}-\mathbf{\lambda})-\mathbf{ \xi})=\widetilde{\mathbf{\alpha}}^{\prime}\widetilde{\mathbf{\omega}}^{-1}(\mathbf{w}- \widetilde{\mathbf{\xi}})\), we obtain that  
\[f_{\mathbf{W}}(\mathbf{w})=\frac{t_{p}(\mathbf{w};\widetilde{\mathbf{\xi}}, \widetilde{\mathbf{\Omega}},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\widetilde {\mathbf{\Omega}}}(\mathbf{w},\widetilde{\mathbf{\xi}})}\Big{]}^{1/2}\widetilde{\mathbf{\alpha }}^{\prime}\widetilde{\mathbf{\omega}}^{-1}(\mathbf{w}-\widetilde{\mathbf{\xi}});\
u+p \Big{)}}{\int_{T(A)}t_{p}(\mathbf{w};\widetilde{\mathbf{\xi}}, \widetilde{\mathbf{\Omega}},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\widetilde {\mathbf{\Omega}}}(\mathbf{w},\widetilde{\mathbf{\xi}})}\Big{]}^{1/2}\widetilde{\mathbf{\alpha }}^{\prime}\widetilde{\mathbf{\omega}}^{-1}(\mathbf{w}-\widetilde{\mathbf{\xi}});\
u+p \Big{)}\,\mathrm{d}\mathbf{w}},\]  
which shows that \(T(\mathbf{Y})\sim\mathrm{TST}_{p}(\widetilde{\mathbf{\xi}},\widetilde{\mathbf{\Omega}}, \widetilde{\mathbf{\alpha}},\
u;T(A))\). **Corollary 1**. : _Let \(T:\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}\) be the transformation \(T(\mathbf{x})=\mathbf{\lambda}+\mathbf{\Delta x}\), being \(\mathbf{\lambda}\in\mathbb{R}^{p}\) a constant vector and \(\mathbf{\Delta}(p\times p)\) a non-singular constant matrix. If \(\mathbf{Y}\sim\mathrm{TT}_{p}(\mathbf{\xi},\mathbf{\Omega},\
u;A)\), then \(T(\mathbf{Y})\sim\mathrm{TT}_{p}(\mathbf{\lambda}+\mathbf{\Delta\xi},\mathbf{\Delta\Omega}^{ \prime},\
u;T(A))\)._  
Proof. : Substitute \(\mathbf{\alpha}=\mathbf{0}\) in Theorem 1.