## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
\begin{array}{l}\ell(\textbf{B},\psi,v)=\ln L(\textbf{B},\psi,v)=\\ \\ =n\ln\!\left[\frac{\Gamma(\frac{v+q}{2})}{(\pi v)^{\frac{q}{2}}\Gamma\big{(} \frac{v}{2}\big{)}|\psi|^{\frac{1}{2}}}\right]-\frac{v+q}{2}\sum\limits_{u=1} ^{n}\ln\!\left[1+\frac{\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{ g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{ \prime}}\textbf{g}(\textbf{x}_{u})\big{)}}{v}\right]\right]\end{array}  \tag{8}\]  
The first partial derivative of the log-likelihood function with respect to each parameter (assumed \(v\) to be known) and equated to zero, i.e., \(\frac{\partial\ell(\textbf{B},\psi)}{\partial\textbf{B}}=\textbf{0}\) and \(\frac{\partial\ell(\textbf{B},\psi)}{\partial\psi}=\textbf{0}\). \[ \begin{array}{l}\frac{\partial\ell(\textbf{B},\psi)}{\partial\textbf{B}}=\\ \\ \frac{\partial\bigg{(}n\ln\!\left[\frac{\frac{\Gamma\big{(}\frac{v+q}{2}\big{)} }}{(\pi v)^{\frac{q}{2}}\Gamma\big{(}\frac{v}{2}\big{)}|\psi|^{\frac{1}{2}}} \right]-\frac{v+q}{2}\sum\limits_{u=1}^{n}\ln\!\left[1\!+\!\frac{\big{(}\textbf {y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi ^{-1}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}_{u })\big{)}}{v}\right]\right)}{\partial\textbf{B}}\\ \\ =-\frac{v+q}{2}\sum\limits_{u=1}^{n}\frac{-2\big{(}\textbf{y}_{u}-\textbf{B}^{{} ^{\prime}}\textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\textbf{g}( \textbf{x}_{u})/v}{\big{[}1+\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\big{(}\textbf{y}_{u}- \textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x}_{u})\big{)}/v\big{]}}\\ \\ =v+q\sum\limits_{u=1}^{n}\frac{\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\textbf{g}(\textbf{x}_{u})}{ \big{[}\psi+\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{g}(\textbf{x }_{u})\big{)}^{\prime}\psi^{-1}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}/v\big{]}}\\ \\ =v+q\sum\limits_{u=1}^{n}\frac{\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\textbf{g}(\textbf{x}_{u})}{ \big{[}\psi+D_{i}(\textbf{B},\psi)\big{]}}\\ \\ =\sum\limits_{u=1}^{n}\frac{(\psi+q)\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\textbf{g}(\textbf{x}_{u})}{ \big{[}\psi+D_{i}(\textbf{B},\psi)\big{]}}\\ \\ =\sum\limits_{u=1}^{n}\omega_{u}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\textbf{g}(\textbf{x}_{u}) \end{array}  \tag{9}\]  
with \(\omega_{u}=\frac{v+q}{v+D_{u}(\textbf{B},\psi)}\) and \(D_{u}(\textbf{B},\psi)=\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}}\textbf{ g}(\textbf{x}_{u})\big{)}^{\prime}\psi^{-1}\big{(}\textbf{y}_{u}-\textbf{B}^{{}^{\prime}} \textbf{g}(\textbf{x}_{u})\big{)}\) as Mahalanobis distance. Then  
\((h+1)\)-th iteration \(h=0,1,2,\ldots\), determine the initial value of each parameter, namely \(\overset{\star}{\mathbf{B}}^{(0)}\) and \(\overset{\star}{\mathbf{\psi}}^{(0)}\).