## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
Expectation (E) Step is done by calculating the value of the auxiliary variable, i.e.,  
\[\omega_{u}^{(h+1)}=\frac{v+q}{v+D_{u}^{(h)}(\mathbf{B},\mathbf{\psi})}\quad \text{and}\quad\overset{\star}{\mathbf{Y}}^{(h)}=\overset{\star}{\mathbf{X}} ^{(h)}\]  
with  
\[D_{u}^{(h)}(\mathbf{B},\mathbf{\psi})=\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\left(\overset{ \star}{\mathbf{\psi}}^{(h)}\right)^{-1}\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)\]  
\[\mathbf{\Omega}=diag(\omega_{1},\omega_{2},\ldots,\omega_{n})\]  
Besides that, it also calculates sufficient statistics:  
\[S_{\omega XX}^{(h+1)}=\overset{n}{\underset{u=1}{\sum}}\omega_{u}^{(h+1)}g( \mathbf{x}_{u})g^{\prime}(\mathbf{x}_{u})=\mathbf{X}^{\prime}\mathbf{\Omega} ^{(h+1)}\mathbf{X}\]  
\[S_{\omega XY}^{(h+1)}=\overset{n}{\underset{u=1}{\sum}}\omega_{u}^{(h+1)}g( \mathbf{x}_{u})\left(\overset{\star}{\mathbf{y}}_{u}\right)^{\prime}=\mathbf{ X}^{\prime}\mathbf{\Omega}^{(h+1)}\overset{\star}{\mathbf{Y}}^{(h)}\]  
\[S_{\omega YY}^{(h+1)}=\overset{n}{\underset{u=1}{\sum}}\omega_{u}^{(h+1)} \left(\overset{\star}{\mathbf{Y}}_{u}\right)\left(\overset{\star}{\mathbf{Y }}_{u}\right)^{\prime}+\overset{n}{\underset{u=1}{\sum}}\overset{n}{\underset{ u=1}{\sum}}\overset{\star}{\mathbf{\psi}}_{u}^{(h)}=\left(\overset{\star}{ \mathbf{Y}}\right)^{\prime}\mathbf{\Omega}^{(h+1)}\left(\overset{\star}{ \mathbf{Y}}^{(h)}\right)+\overset{n}{\underset{u=1}{\sum}}\overset{n}{ \underset{u=1}{\sum}}\overset{\star}{\mathbf{\psi}}_{u}^{(h)}\]  
Furthermore, Maximization (M) Step is done to get the estimator value of \(\mathbf{B}\) and \(\mathbf{\Psi}\), namely:  
\[\overset{\star}{\mathbf{B}}^{(h+1)} =\left(S_{\omega XX}^{(h+1)}\right)^{-1}S_{\omega XY}^{(h+1)}= \left(\mathbf{X}^{\prime}\mathbf{\Omega}^{(h+1)}\mathbf{X}\right)^{-1}\mathbf{ X}^{\prime}\mathbf{\Omega}^{(h+1)}\overset{\star}{\mathbf{Y}}^{(h)}\] \[\overset{\star}{\mathbf{Y}}^{(h+1)} =\overset{n}{\underset{u=1}{\sum}}\left[S_{\omega YY}^{(h+1)}- \left(S_{\omega XY}^{(h+1)}\right)^{\prime}\left(S_{\omega XX}^{(h+1)}\right)^ {-1}S_{\omega XY}^{(h+1)}\right] \tag{13}\] \[=\overset{n}{\underset{u=1}{\sum}}\left(\overset{\star}{\mathbf{ Y}}^{(h+1)}-\overset{\star}{\mathbf{X}}^{(h+1)}\right)^{\prime}\mathbf{\Omega}^{(h+1)} \left(\overset{\star}{\mathbf{Y}}^{(h+1)}-\overset{\star}{\mathbf{X}}^{(h+1) }\right)+\overset{n}{\underset{u=1}{\sum}}\overset{n}{\underset{u=1}{ \sum}}\overset{\star}{\mathbf{\psi}}^{(h)}\]  
Variance-covariance matrix for \(\mathbf{B}\) can be obtained through a fisher information matrix for \(\mathbf{B}\)[20], i.e.,  
\[\mathbf{K}(\mathbf{B})=\overset{n}{\underset{u=1}{\sum}}\left(\frac{v+q}{v+q +2}\right)g(\mathbf{x}_{u})\mathbf{\psi}^{-1}g^{\prime}(\mathbf{x}_{u}) \tag{14}\]  
The consistent asymptotic estimator of the variance-covariance matrix for \(\mathbf{B}\) is:  
\[D_{u}^{(h)}(\mathbf{B},\mathbf{\psi})=\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\left(\overset{ \star}{\mathbf{\psi}}^{(h)}\right)^{-1}\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)\]0  
Because the MLE method produces an estimated value that approaches the actual value of the estimated parameter, the \(\overset{\star}{\mathbf{B}}\) is consistent, so that \(\left(\overset{\star}{\mathbf{B}}-\mathbf{B}\right)\) can be approached with a multivariate normal distribution, i.e.,  
\[D_{u}^{(h)}(\mathbf{B},\mathbf{\psi})=\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)^{\prime}\left(\overset{ \star}{\mathbf{\psi}}^{(h)}\right)^{-1}\left(\mathbf{y}_{u}-\overset{\star}{ \mathbf{B}}^{(h)}\mathbf{g}(\mathbf{x}_{u})\right)\]1  
Then, the algorithm for model parameter estimation using EM is as follows:  
The log-likelihood function, i.e.,  
\[