## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
\begin{split}\ell\biggl{(}\overset{\mathbf{\alpha}}{\mathbf{\Theta}} \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^{\mathbf{\alpha}}\bigg{|}+\\ &\quad\quad\quad\quad-\frac{v+q}{2}\sum\limits_{u=1}^{n}\ln\Biggl{[} 1+\frac{\left(\mathbf{y}_{u}-\overset{\mathbf{\alpha}}{\mathbf{B}}^{{}^{\prime}} \mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\Psi_{0}^{-1}\biggl{(}\mathbf{y}_{u }-\overset{\mathbf{\alpha}}{\mathbf{B}}^{{}^{\prime}}\mathbf{g}(\mathbf{x}_{u}) \biggr{)}}{v}\Biggr{]}\Biggr{]}\end{split}  \tag{26}\]where \(\left(\overset{\wedge}{\beta},\overset{\wedge}{\psi}\right)\) is an estimator for \((\beta,\psi)\) on the SOMRS model with multivariate t-distribution as stated in Equation (5). **Theorem 2. **_If \(\overset{\wedge}{\Theta}\) is the parameter set under the population, \(\overset{\wedge}{\Theta}_{0}\) is the parameter set under \(\mathrm{H}_{0}\), and a hypothesis is given by Equation (17), then the test statistic for the hypothesis testing in that Equation is \(\zeta=2\bigg{(}\ell\bigg{(}\overset{\wedge}{\Theta}\bigg{)}-\ell\bigg{(} \overset{\wedge}{\Theta}_{0}\bigg{)}\bigg{)}\)_  
**Proof of Theorem 2. ** The likelihood ratio for hypothesis testing Equation (17):  
\[R=\frac{L\bigg{(}\overset{\wedge}{\Theta}_{0}\bigg{)}}{L\bigg{(}\overset{\wedge }{\Theta}\bigg{)}}<R_{0} \tag{27}\]  
where \(R_{0}\) is a constant whose value lies between 0 and 1. \[R=\frac{\prod\limits_{u=1}^{n}\bigg{|}\overset{\wedge}{\Psi}_{0}\bigg{|}^{- \frac{1}{2}}\Bigg{[}1+\frac{\bigg{(}\mathbf{y}_{u}-\overset{\wedge}{\Theta}_ {0}\bigg{)}^{\prime}\Psi_{0}^{-1}\bigg{(}\mathbf{y}_{u}-\overset{\wedge}{ \Theta}_{0}\bigg{)}}{v}\Bigg{]}^{-\frac{p+q}{2}}}{\prod\limits_{u-1}^{n} \bigg{|}\overset{\wedge}{\Phi}\bigg{|}^{-\frac{1}{2}}\Bigg{[}1+\frac{\bigg{(} \mathbf{y}_{u}-\overset{\wedge}{\mathbf{B}}_{0}\mathbf{g}(\mathbf{x}_{u}) \bigg{)}^{\prime}\Psi_{0}^{-1}\bigg{(}\mathbf{y}_{u}-\overset{\wedge}{ \mathbf{B}}_{0}\mathbf{g}(\mathbf{x}_{u})\bigg{)}}{v}\Bigg{]}^{-\frac{p+q}{2}}} \Bigg{]}^{-\frac{p+q}{2}} \tag{28}\]  
Based on Equation (28), the R-value is difficult to simplify, so to simplify calculations, the likelihood ratio in Equation (27) is expressed in an equivalent form, namely  
\[R^{2}=\Bigg{[}\frac{L\bigg{(}\overset{\wedge}{\Theta}_{0}\bigg{)}}{L\bigg{(} \overset{\wedge}{\Theta}\bigg{)}}\Bigg{]}^{-2}=\Bigg{[}\frac{L\bigg{(}\overset {\wedge}{\Theta}\bigg{)}}{L\bigg{(}\overset{\wedge}{\Theta}_{0}\bigg{)}} \Bigg{]}^{2} \tag{29}\]  
Applying the natural logarithm to the Equation (29) yields the test statistic  
\[\zeta =\ln\big{(}R^{-2}\big{)}=-2\ln(R)=-2\ln\Bigg{[}\frac{L\big{(} \overset{\wedge}{\Theta}_{0}\big{)}}{L\big{(}\overset{\wedge}{\Theta}\big{)} }\Bigg{]}=2\ln\Bigg{[}\frac{L\big{(}\overset{\wedge}{\Theta}\big{)}}{L\big{(} \overset{\wedge}{\Theta}_{0}\big{)}}\Bigg{]} \tag{30}\] \[=2\bigg{(}\ell\bigg{(}\overset{\wedge}{\Theta}\bigg{)}-\ell \bigg{(}\overset{\wedge}{\Theta}_{0}\bigg{)}\bigg{)}\]  
with \(\ell\bigg{(}\overset{\wedge}{\Theta}\bigg{)}\) according to Equation (26) and \(\ell\bigg{(}\overset{\wedge}{\Theta}_{0}\bigg{)}\) according to Equation (22).\(\square\)  
**Theorem 3. **_Based on Theorem 2, the distribution of test statistics \(\zeta\) is Chi-square with the degrees of freedom \(kq\), which can be written as follows:_  
\[\zeta=2\bigg{(}\ell\bigg{(}\overset{\wedge}{\Theta}\bigg{)}-\ell\bigg{(} \overset{\wedge}{\Theta}_{0}\bigg{)}\bigg{)}\overset{d}{\rightarrow}\chi_{ kq}^{2},k\rightarrow\infty \tag{31}\]  
By substituting the Equations (34) and (36) to Equation (32) obtained  
\[