# Some Statistical Aspects of the Truncated Multivariate Skew-\(t\) Distribution
## 2 The Truncated Multivariate Skew-\(t\) Distribution
The random vector \(\mathbf{Y}\in A\) has a truncated multivariate skew-\(t\) distribution with support \(A\) and parameters \(\mathbf{\xi}\in\mathbb{R}^{p}\), \(\mathbf{\Omega}(p\times p)>0\), \(\mathbf{\alpha}\in\mathbb{R}^{p}\) and \(\
u>0\) degrees of freedom, denoted by \(\mathbf{Y}\sim\mathrm{ST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;A)\), if its PDF is given by  
\[f_{\mathbf{Y}}(\mathbf{y})=\frac{t_{p}(\mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[} \frac{\
u+p}{\
u+\delta_{\mathbf{\Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{ \alpha}^{\prime}\mathbf{\omega}^{-1}(\mathbf{y}-\mathbf{\xi});\
u+p\Big{)}}{\int_{A}t_{p} (\mathbf{y};\mathbf{\xi},\mathbf{\Omega},\
u)T\Big{(}\Big{[}\frac{\
u+p}{\
u+\delta_{\mathbf{ \Omega}}(\mathbf{y},\mathbf{\xi})}\Big{]}^{1/2}\mathbf{\alpha}^{\prime}\mathbf{\omega}^{-1}( \mathbf{y}-\mathbf{\xi});\
u+p\Big{)}\,\mathrm{d}\mathbf{y}},\quad\mathbf{y}\in A. \tag{7}\]  
The PDF of \(\mathbf{Y}\sim\mathrm{TST}_{p}(\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u;A)\) can be expressed in equivalent form as  
\[f_{\mathbf{Y}}(\mathbf{y})=\frac{\mathrm{ST}_{p}(\mathbf{y};\mathbf{\xi},\mathbf{\Omega},\mathbf{ \alpha},\
u)}{\mathrm{ST}_{p}(A;\mathbf{\xi},\mathbf{\Omega},\mathbf{\alpha},\
u)},\quad \mathbf{y}\in A, \tag{8}\]where \(\text{ST}_{p}(A;\xi,\Omega,\alpha,\
u)=\int_{A}\text{ST}_{p}(\mathbf{y};\xi,\Omega, \alpha,\
u)\,\text{d}\mathbf{y}\), being \(\text{ST}_{p}(\mathbf{x};\xi,\Omega,\alpha,\
u)\) the PDF of \(\mathbf{X}\), \(\mathbf{X}\sim\text{ST}_{p}(\xi,\Omega,\alpha,\
u)\), given in (2). Note that if \(A=\mathbb{R}^{p}\) in (8), we obtain the multivariate skew-\(t\) distribution (Definition 1) as a particular case of the truncated multivariate skew-\(t\) distribution. If we take \(\mathbf{\alpha}=\mathbf{0}\) in (7), then we obtain the PDF of a random vector \(Y\in A\) with truncated multivariate \(t\) distribution with support \(A\) and parameters \(\xi\in\mathbb{R}^{p}\), \(\Omega(p\times p)>0\) and \(\
u>0\) degrees of freedom, which we denote by \(\mathbf{Y}\sim\text{TT}_{p}(\xi,\Omega,\
u;A)\) ([PERSON] and [PERSON] [8]). As expected, the truncated multivariate skew-normal distribution ([PERSON] et al. [9]) is obtained as a limiting case of the truncated multivariate skew-\(t\) distribution when \(\
u\to\infty\). Additional special cases of the truncated multivariate skew-\(t\) distribution are the multivariate \(t\) distribution (\(A=\mathbb{R}^{p}\) and \(\mathbf{\alpha}=\mathbf{0}\) in (8)) and the multivariate normal distribution (\(A=\mathbb{R}^{p}\), \(\mathbf{\alpha}=\mathbf{0}\) and \(\
u\to\infty\) in (8)). Figure 1 exhibit different shapes of the truncated bivariate skew-\(t\) distribution with rectangular support according to its parameters. We consider different parameter settings and plot the surface of the PDF (7) and its contours for different levels, namely \(f_{\mathbf{Y}}(\mathbf{y})=k\), where \(k=0.33,0.25,0.14,0.06,0.02,0.006,0.002\). The legend indicates the values of all the parameters considered in the first plot and the values of the parameters that are changed from a plot to the subsequent one (in alphabetical order). In Figure 1a, where \(\mathbf{\alpha}=\mathbf{0}\), the contours are truncated ellipses. When \(\mathbf{\alpha}\
eq\mathbf{0}\), these ellipses are deformed (Figure 1b). Figure 1c shows the effect of parameter \(\omega_{12}\). As the degrees of freedom parameter grows, the contours of the truncated bivariate skew-\(t\) distribution tend to the contours of truncated bivariate skew-normal distribution (Figure 1c-d). The tails of the truncated bivariate skew-\(t\) distribution are heavier for smaller values of \(\
u\). The truncated multivariate skew-\(t\) distribution is appropriate for modeling multivariate data whose values are restricted to a subset of \(\mathbb{R}^{p}\), and they are possibly skewed and heavy-tailed. This type of data occurs in a variety of situations. For instance, the dataset studied in [PERSON] et al.