# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 3 Distributional Properties
Thus we obtain  
\[\int_{\mathbb{R}^{p_{2}}}\phi_{p_{2}}(\mathbf{y}_{2};\mathbf{\xi}_{2\cdot 1},u^{-1}\mathbf{ \Omega}_{22:1})\Phi(\sqrt{u}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}-\mathbf{\xi}))\, \mathrm{d}\mathbf{y}_{2}=(2\pi)^{-p_{2}/2}\det(u^{-1}\mathbf{\Omega}_{22:1})^{-1/2}\] \[\times\int_{\mathbb{R}^{p_{2}}}\exp\Bigl{(}-\frac{u\mathbf{x}^{ \prime}\mathbf{\Omega}_{22:1}^{-1}\mathbf{x}}{2}\Bigr{)}\Phi(\sqrt{u}[(1+\mathbf{a}_{2}^{ \prime}\mathbf{\omega}_{2}^{-1}\mathbf{\Omega}_{22:1}\mathbf{\omega}_{2}^{-1}\mathbf{\alpha}_{2})^{ 1/2}\mathbf{a}_{1(2)}^{\prime}\mathbf{\omega}_{1}^{-1}(\mathbf{y}_{1}-\mathbf{\xi}_{1})+\mathbf{a}_{2 }^{\prime}\mathbf{\omega}_{2}^{-1}\mathbf{x}])\,\mathrm{d}\mathbf{x}\] \[=\Phi(\sqrt{u}\mathbf{\alpha}_{1(2)}^{\prime}\mathbf{\omega}_{1}^{-1}( \mathbf{y}_{1}-\mathbf{\xi}_{1})),\]where the last line is derived by using Lemma 2 with \(\mathbf{G}=u^{-1}\mathbf{\Omega}_{22\cdot 1}\), \(\mathbf{b}=\mathbf{0}\), \(\mathbf{c}=\sqrt{u}\omega_{2}^{-1}\mathbf{a}_{2}\) and \(k=\sqrt{u}(1+\mathbf{\alpha}_{2}^{\prime}\mathbf{\omega}_{2}^{-1}\mathbf{\Omega}_{22\cdot 1} \mathbf{\omega}_{2}^{-1}\mathbf{\alpha}_{2})^{1/2}\mathbf{\alpha}_{1(2)}^{\prime}\mathbf{\omega} _{1}^{-1}(\mathbf{y}_{1}-\mathbf{\xi}_{1})\). Therefore, in (13) we have  
\[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1}) =\frac{\int_{0}^{\infty}\phi_{p_{1}}(\mathbf{y}_{1};\mathbf{\xi}_{1},u^{ -1}\mathbf{\Omega}_{11})\Phi(\sqrt{u}\mathbf{\alpha}_{1(2)}^{\prime}\mathbf{\omega}_{1}^{- 1}(\mathbf{y}_{1}-\mathbf{\xi}_{1}))h(u;\
u)\,\mathrm{d}u}{\int_{R_{1}}\int_{0}^{ \infty}\phi_{p_{1}}(\mathbf{y}_{1};\mathbf{\xi}_{1},u^{-1}\mathbf{\Omega}_{11})\Phi(\sqrt{ u}\mathbf{\alpha}_{1(2)}^{\prime}\mathbf{\omega}_{1}^{-1}(\mathbf{y}_{1}-\mathbf{\xi}_{1}))h(u; \
u)\,\mathrm{d}u\,\mathrm{d}\mathbf{y}_{1}}\] \[f_{\mathbf{Y}_{1}}(\mathbf{y}_{1})=\frac{\int_{\mathbb{R}^{p}_{2}}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}_{2}}{\int_{R}t_{p}( \mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\omega^{-1}(\mathbf{y}- \mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}}, \tag{11}\]0  
where the last line is obtained by using (12). This completes the proof. Corollary 2: _Let \(\mathbf{Y}\sim\mathrm{TT}_{p}(\mathbf{\xi},\mathbf{\Omega},\
u;R)\). Consider the partitions given in (9) and (10). If \(R_{2}=\mathbb{R}^{p_{2}}\), then \(\mathbf{Y}_{1}\sim\mathrm{TT}_{p_{1}}(\mathbf{\xi}_{1},\mathbf{\Omega}_{11},\
u;R_{1})\)._  
Proof. : Substitute \(\mathbf{\alpha}=\mathbf{0}\) in Theorem 2. From Theorem 3.7 of [PERSON] and Ferrari [8], we can establish that if \(\mathbf{\Omega}_{12}=\mathbf{0}\), then the marginals of the truncated multivariate \(t\) distribution belong to the class of truncated elliptical distributions (but do not necessarily belong to the truncated multivariate \(t\) family). Corollary 2 allows us to establish closedness of the truncated multivariate \(t\) distribution for some marginals by restricting the support set without constraining the values of \(\mathbf{\Omega}\). In Definition 4, we present the truncated multivariate extended skew-\(t\) distribution, which is needed to derive the conditional distribution of \(\mathbf{Y}_{2}|\mathbf{Y}_{1}\), where \(\mathbf{Y}=(\mathbf{Y}_{1}^{\prime},\mathbf{Y}_{2}^{\prime})^{\prime}\) has a truncated multivariate skew-\(t\) distribution. The truncated multivariate extended skew-\(t\) distribution is obtained by conditioning \(\mathbf{X}\sim\mathrm{EST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\tau,\
u)\) (Definition 2) on \(\{\mathbf{X}\in A\}\), where \(A\subseteq\mathbb{R}^{p}\) is a measurable set. Definition 4: Let \(A\subseteq\mathbb{R}^{p}\) be a measurable set.