# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 2 The Truncated Multivariate Skew-\(t\) Distribution  
In what follows, if \(\mathbf{\Omega}\) is a square matrix, then \(\det(\mathbf{\Omega})\) denotes the determinant of \(\mathbf{\Omega}\). If \(\mathbf{\Omega}\) is a symmetric matrix, then \(\mathbf{\Omega}>0\) means that \(\mathbf{\Omega}\) is positive definite. Furthermore, \(\mathbf{\Omega}^{1/2}\) denotes the unique symmetric positive definite square root of \(\mathbf{\Omega}>0\). It is well-known that a random vector \(\mathbf{W}\in\mathbb{R}^{p}\) has a multivariate \(t\) distribution with location vector \(\mathbf{\xi}\in\mathbb{R}^{p}\), dispersion matrix \(\mathbf{\Omega}(p\times p)>0\), and degrees of freedom \(\
u>0\), if its probability density function (PDF) is  
\[t_{p}(\mathbf{w};\mathbf{\xi},\mathbf{\Omega},\
u)=\frac{\Gamma[(\
u+p)/2]}{\det( \mathbf{\Omega})^{1/2}(\pi\
u)^{p/2}\Gamma(\
u/2)}\bigg{(}1+\frac{\delta_{ \mathbf{\Omega}}(\mathbf{w},\mathbf{\xi})}{\
u}\bigg{)}^{-(\
u+p)/2},\quad\mathbf{w}\in \mathbb{R}^{p}, \tag{1}\]  
where \(\delta_{\mathbf{\Omega}}(\mathbf{w},\mathbf{\xi})=(\mathbf{w}-\mathbf{\xi})^{\prime}\mathbf{ \Omega}^{-1}(\mathbf{w}-\mathbf{\xi})\) is the square of the Mahalanobis distance between \(\mathbf{w}\) and \(\mathbf{\xi}\) with respect to \(\mathbf{\Omega}\). When \(\
u\to\infty\) in (1), we obtain the PDF of the multivariate normal vector \(\mathbf{W}\sim\mathrm{N}_{p}(\mathbf{\xi},\mathbf{\Omega})\), which is given by \(\phi_{p}(\mathbf{w};\mathbf{\xi},\mathbf{\Omega})=(2\pi)^{-p/2}\det(\mathbf{\Omega})^ {-1/2}\exp[-\delta_{\mathbf{\Omega}}(\mathbf{w},\mathbf{\xi})/2]\). In addition, if \(\
u=1\) in (1), we retrieve the PDF of the multivariate Cauchy distribution. A detailed study on the multivariate \(t\) distribution appear in [PERSON] and [PERSON] [26]. The PDF of the multivariate skew-\(t\) distribution can be expressed in terms of the PDF (1) as in Definition 1 ([PERSON] and [PERSON] [3]). Definition 1: The random vector \(\mathbf{X}\in\mathbb{R}^{p}\) is said to have a multivariate skew-\(t\) distribution with location vector \(\mathbf{\xi}\in\mathbb{R}^{p}\), dispersion matrix \(\mathbf{\Omega}(p\times p)>0\), shape parameter \(\mathbf{\alpha}\in\mathbb{R}^{p}\) and \(\
u>0\) degrees of freedom, denoted by \(\mathbf{X}\sim\mathrm{ST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u)\), if its PDF is  
\[\mathrm{ST}_{p}(\mathbf{x};\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u)=2t_{p}(\mathbf{x} ;\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{x},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\mathbf{\omega}^{-1}( \mathbf{x}-\mathbf{\xi});\
u+p\Big{)}, \tag{2}\]  
where \(\mathbf{x}\in\mathbb{R}^{p}\), \(\mathbf{\omega}=(\mathbf{\Omega}\odot\mathbf{I}_{p})^{1/2}\), with \(\odot\) being the Hadamard product, and \(T(z;\
u)\) is the CDF of a standard \(t\) random variable with \(\
u>0\) degrees of freedom. The multivariate skew-normal distribution is obtained as a limiting case of the multivariate skew-\(t\) distribution when \(\
u\to\infty\), with the PDF given by  
\[\mathrm{SN}_{p}(\mathbf{x};\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha})=2\phi_{p}(\mathbf{x};\mathbf{\xi },\mathbf{\Omega})\Phi(\mathbf{\alpha}^{\prime}\mathbf{\omega}^{-1}(\mathbf{x}-\mathbf{\xi})),\quad \mathbf{x}\in\mathbb{R}^{p}, \tag{3}\]  
where \(\Phi(z)\) is the CDF of a standard normal random variable. We write \(\mathbf{X}\sim\mathrm{SN}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha})\) if the PDF of \(\mathbf{X}\) is given by (3). In addition, the multivariate \(t\) distribution is obtained from the multivariate skew-\(t\) distribution when \(\mathbf{\alpha}=0=(0,\ldots,0)^{\prime}\). A stochastic representation for \(\mathbf{X}\sim\mathrm{ST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u)\) is \(\mathbf{X}|U=u\sim\mathrm{SN}_{p}(\mathbf{\xi},u^{-1}\mathbf{\Omega},\mathbf{\alpha})\), \(U\sim\mathrm{Gamma}(\
u/2,\
u/2)\); see [PERSON] and Genton [3].