## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
\begin{split}&\ell(\mathbf{\Theta}_{0})=\ln L(\mathbf{\Theta}_{0})=\sum \limits_{u=1}^{n}f(\mathbf{y}_{u}|\mathbf{B}_{0},\Psi_{0})\\ &=n\ln\Biggl{[}\frac{\Gamma(\frac{v+q}{2})}{(\pi v)^{\frac{q}{2}} \Gamma(\frac{v}{2})|\Psi_{0}|^{\frac{1}{2}}}\Biggr{]}-\frac{v+q}{2}\sum \limits_{u=1}^{n}\ln\Biggl{[}1+\frac{(\mathbf{y}_{u}-\mathbf{B}^{{}^{\prime}} \mathbf{g}(\mathbf{x}_{u}))^{\prime}\Psi_{0}^{-1}(\mathbf{y}_{u}-\mathbf{B}^{{ }^{\prime}}\mathbf{g}(\mathbf{x}_{u}))}{v}\Biggr{]};\end{split}  \tag{20}\]  
So that the maximum likelihood function under H\({}_{0}\) is  
\[L\biggl{(}\overset{\mathbf{\alpha}}{\mathbf{\Theta}}_{0}\biggr{)}=\frac{\left[\Gamma \left(\frac{v+q}{2}\right)\right]^{n}}{(\pi v)^{\frac{q}{2}}\left[\Gamma\left( \frac{v}{2}\right)\right]^{n}}\cdot\prod\limits_{u=1}^{n}\bigg{|}\psi_{0}^{- 1}\Biggl{[}1+\frac{\left(\mathbf{y}_{u}-\overset{\mathbf{\alpha}}{\mathbf{\beta}}^{{}^ {\prime}}_{0}\right)^{\prime}\Psi_{0}^{-1}\Biggl{(}\mathbf{y}_{u}-\overset{ \mathbf{\alpha}}{\mathbf{\beta}}^{{}^{\prime}}_{0}\Biggr{)}}{v}\Biggr{]}^{-\frac{v+q} {2}} \tag{21}\]  
and the maximum of log-likelihood function under H\({}_{0}\) is  
\[\ell\biggl{(}\overset{\mathbf{\alpha}}{\mathbf{\Theta}}_{0}\biggr{)}=n\ln\Biggl{[} \frac{\Gamma\left(\frac{v+q}{2}\right)}{(\pi v)^{\frac{q}{2}}\Gamma\left( \frac{v}{2}\right)}\Biggr{]}-\frac{1}{2}\sum\limits_{u=1}^{n}\ln\biggl{|}\psi_ {0}\biggr{|}-\frac{v+q}{2}\sum\limits_{u=1}^{n}\ln\Biggl{[}1+\frac{\left( \mathbf{y}_{u}-\overset{\mathbf{\alpha}}{\mathbf{\beta}}^{{}^{\prime}}_{0}\right)^{ \prime}\Psi_{0}^{-1}\Biggl{(}\mathbf{y}_{u}-\overset{\mathbf{\alpha}}{\mathbf{\beta}} ^{{}^{\prime}}_{0}\Biggr{)}}{v}\Biggr{]} \tag{22}\]  
where \(\biggl{(}\overset{\mathbf{\alpha}}{\mathbf{\beta}}_{0},\overset{\mathbf{\alpha}}{\mathbf{ \beta}}_{0}\biggr{)}\) is an estimator for \((\beta_{0},\Psi_{0})\) on the SOMRS model with a multivariate t-distribution approach as stated in Equation (5). Next is to determine the likelihood function under the population, i.e.,  
\[L(\mathbf{\Theta})=\frac{\left[\Gamma\left(\frac{v+q}{2}\right)\right]^{n}}{(\pi v )^{\frac{q}{2}}\left[\Gamma\left(\frac{v}{2}\right)\right]^{n}|\Psi|^{\frac{1 }{2}}}\cdot\prod\limits_{u=1}^{n}\Biggl{[}1+\frac{\left(\mathbf{y}_{u}- \mathbf{B}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\Psi^{-1} \bigl{(}\mathbf{y}_{u}-\mathbf{B}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u}) \bigr{)}}{v}\Biggr{]}^{-\frac{v+q}{2}} \tag{23}\]  
Then, the log-likelihood function under the population is  
\[\ell(\mathbf{\Theta})=n\ln\Biggl{[}\frac{\Gamma\left(\frac{v+q}{2}\right)}{(\pi v )^{\frac{q}{2}}\Gamma\left(\frac{v}{2}\right)|\Psi|^{\frac{1}{2}}}\Biggr{]}- \frac{v+q}{2}\sum\limits_{u=1}^{n}\ln\Biggl{[}1+\frac{\left(\mathbf{y}_{u}- \mathbf{B}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\Psi^{-1} \bigl{(}\mathbf{y}_{u}-\mathbf{B}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u}) \bigr{)}}{v}\Biggr{]} \tag{24}\]  
So, the maximum of likelihood function under the population is  
\[L\biggl{(}\overset{\mathbf{\alpha}}{\mathbf{\Theta}}\biggr{)}=\frac{\left[\Gamma\left( \frac{v+q}{2}\right)\right]^{n}}{(\pi v)^{\frac{q}{2}}\left[\Gamma\left(\frac {v}{2}\right)\right]^{n}}\cdot\prod\limits_{u=1}^{n}\bigg{|}\psi^{\mathbf{\alpha} }\bigg{|}^{-\frac{1}{2}}\Biggl{[}1+\frac{\left(\mathbf{y}_{u}-\overset{\mathbf{ \alpha}}{\mathbf{B}}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime }\overset{\mathbf{\alpha}}{\mathbf{\psi}}^{-1}\Biggl{(}\mathbf{y}_{u}-\overset{\mathbf{ \alpha}}{\mathbf{B}}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u})\Biggr{)}}{v} \Biggr{]}^{-\frac{v+q}{2}} \tag{25}\]  
Moreover, the maximum of log-likelihood function under the population is  
\[