## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
\begin{split}&\zeta=2\Big{(}\ell\left(\overset{\wedge}{\mathbf{ \Theta}}\right)-\ell\left(\overset{\wedge}{\mathbf{\Theta}}_{0}\right)\Big{)} \\ &=\left(\overset{\wedge}{\mathbf{\Theta}}_{1}-\overset{\wedge}{ \mathbf{\Theta}}_{10}\right)^{\prime}\left[\mathbf{K}^{11}\left(\overset{ \wedge}{\mathbf{\Theta}}\right)\right]^{-1}\left(\overset{\wedge}{\mathbf{ \Theta}}_{1}-\overset{\wedge}{\mathbf{\Theta}}_{10}\right)\\ &=\mathbf{\beta}^{*\prime}\left[\mathbf{K}^{11}\left(\overset{\wedge}{ \mathbf{\Theta}}\right)\right]^{-1}\mathbf{\beta}^{*}\\ &=\left[\left[\mathbf{K}^{11}\left(\overset{\wedge}{\mathbf{\Theta}}\right) \right]^{-\frac{1}{2}}\mathbf{\beta}^{*\prime}\right]^{\prime}\left[\left[ \mathbf{K}^{11}\left(\overset{\wedge}{\mathbf{\Theta}}\right)\right]^{-\frac{1}{ 2}}\mathbf{\beta}^{*\prime}\right]\\ &=\overset{\wedge}{\mathbf{z}}\overset{d}{\mathbf{z}}\overset{d}{ \rightarrow}\chi_{(kq)}^{2},k\rightarrow\infty\end{split}  \tag{51}\]  
The critical area for the hypothesis testing of the model parameters simultaneously is  
\[ \begin{split}&\alpha=P(R<R_{0})\\ &=P(-2\ln R>-2\ln R_{0})\\ &=P(\zeta>c)\end{split} \]where \(c=-2\ln R_{0}\), so that the following is obtained:  
\[\alpha=P\Big{(}\zeta>\chi_{\alpha,kq}^{2}\Big{)} \tag{52}\]  
Based on Equation (52), the critical area for the simultaneous hypothesis testing of the SOMRS model with multivariate t-distribution is to reject H\({}_{0}\) if the value is \(\zeta>\chi_{\alpha,kq}^{2}\). Then, the algorithm for hypothesis testing of the SOMRS model is as follows:  
* Determine the estimator value for the set of model parameters under H\({}_{0}\) and under the population by running the EM algorithm;
* Determine the null hypothesis (H\({}_{0}\)) and alternative hypotheses (H\({}_{1}\)) \[ \begin{array}{l}\text{H}_{0}:\beta_{1i}=\ldots=\beta_{ki}=\beta_{11i}=\ldots =\beta_{kki}=\beta_{12i}=\ldots=\beta_{(k-1)ki}=0\\ \text{H}_{1}:\text{Atleastoneof}\beta_{ji}\
eq\text{0 ata}\beta_{jji}\
eq\text{ 0 or}\beta_{(j-1)ji}\
eq 0,i=1,2,\ldots,q,\\ j=1,2,\ldots,k\end{array} \]
* Calculate the value of \(\ell\Big{(}\overleftarrow{\boldsymbol{\Theta}}_{0}\Big{)}\), i.e., \[\ell\Big{(}\overleftarrow{\boldsymbol{\Theta}}_{0}\Big{)}=n\ln\left[\frac{ \Gamma\Big{(}\frac{v+q}{2}\Big{)}}{(\pi v)^{\frac{q}{2}}\Gamma\big{(}\frac{v} {2}\Big{)}}\right]-\frac{1}{2}\sum\limits_{u=1}^{n}\ln\left[1+\frac{\left( \boldsymbol{y}_{u}-\overleftarrow{\boldsymbol{\beta}^{{}^{\prime}}_{0}} \right)^{\prime}\boldsymbol{\psi}_{0}^{-1}\bigg{(}\boldsymbol{y}_{u}- \overleftarrow{\boldsymbol{\beta}^{{}^{\prime}}_{0}}\bigg{)}}{v}\right]\] Step-4: Calculate the value of \(\ell\Big{(}\overleftarrow{\boldsymbol{\Theta}}\Big{)}\), i.e., \[\ell\Big{(}\overleftarrow{\boldsymbol{\Theta}}\Big{)}= n\ln\left[\frac{\Gamma\Big{(}\frac{v+q}{2}\Big{)}}{(\pi v)^{\frac{q}{2}}\Gamma\big{(}\frac{v}{2}\Big{)}}\right]-\frac{1}{2} \sum\limits_{u=1}^{n}\ln\left|\boldsymbol{\psi}\right|+\] \[-\frac{v+q}{2}\sum\limits_{u=1}^{n}\ln\left[1+\frac{\left( \boldsymbol{y}_{u}-\overleftarrow{\boldsymbol{\beta}^{{}^{\prime}}_{0}} \boldsymbol{\mathrm{(x_{u})}}\right)^{\prime}\boldsymbol{\psi}_{0}^{-1}\bigg{(} \boldsymbol{y}_{u}-\overleftarrow{\boldsymbol{\beta}^{{}^{\prime}}_{0}} \boldsymbol{\mathrm{(x_{u})}}\bigg{)}}{v}\right]\] Step-5: Calculate the value of the statistical test \(\zeta=2\Big{(}\ell\Big{(}\overleftarrow{\boldsymbol{\Theta}}\Big{)}-\ell \Big{(}\overleftarrow{\boldsymbol{\Theta}}_{0}\Big{)}\Big{)}\); Step-6: Determine the level of significance \(\alpha\) and critical value \(\chi_{\alpha,kq}^{2}\); Step-7: Determine the rejection area, i.e., reject H\({}_{0}\) if \(\zeta>\chi_{\alpha,kq}^{2}\); Step-8: Make conclusions and interpretations.