## 3 Results
### Parameter Estimation of SOMRS Model with a Multivariate \(t\)-Distribution Approach
\begin{array}{ll}\zeta&=\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}- \boldsymbol{\Theta}_{0}\bigg{)}^{\prime}\bigg{[}\mathbf{K}\bigg{(}\overset{ \wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]}\bigg{(}\overset{\wedge}{ \boldsymbol{\Theta}}-\boldsymbol{\Theta}_{0}\bigg{)}-\bigg{(}\overset{\wedge}{ \boldsymbol{\Theta}}_{0}-\boldsymbol{\Theta}_{0}\bigg{)}^{\prime}\bigg{[} \mathbf{K}\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]}\bigg{(} \overset{\wedge}{\boldsymbol{\Theta}}_{0}-\boldsymbol{\Theta}_{0}\bigg{)}\\ &=\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}-\overset{\wedge}{\boldsymbol{ \Theta}}_{0}\bigg{)}^{\prime}\bigg{[}\mathbf{K}\bigg{(}\overset{\wedge}{ \boldsymbol{\Theta}}\bigg{)}\bigg{]}\bigg{(}\overset{\wedge}{\boldsymbol{ \Theta}}-\overset{\wedge}{\boldsymbol{\Theta}}_{0}\bigg{)}\end{array}  \tag{37}\]  
The Fisher Information matrix and its inverse in the Equation (37) can be stated as follows  
\[\bigg{[}\mathbf{K}\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]} = \begin{bmatrix}[\mathbf{K}_{11}]&[\mathbf{K}_{12}]\\ [\mathbf{K}_{21}]&[\mathbf{K}_{22}]\end{bmatrix}  \tag{38}\]  
and  
\[\bigg{[}\mathbf{K}\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]} ^{-1}= \begin{bmatrix}[\mathbf{K}^{11}]&[\mathbf{K}^{12}]\\ [\mathbf{K}^{21}]&[\mathbf{K}^{22}]\end{bmatrix}  \tag{39}\]  
The inverse of the Fisher Information Matrix in the Equation (39) can be declared to be  
\[\bigg{[}\mathbf{K}\bigg{(}\overset{\wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]} ^{-1}= \begin{bmatrix}[\mathbf{K}^{-1}_{11,2}]&-\Big{[}\mathbf{K}^{-1}_{11,2} \Big{]}[\mathbf{K}_{12}]\Big{[}\mathbf{K}^{-1}_{22}\Big{]}\\ -\Big{[}\mathbf{K}^{-1}_{22}\Big{]}[\mathbf{K}_{21}]\Big{[}\mathbf{K}^{-1}_{11,2}\Big{]}&\Big{[}\mathbf{K}^{-1}_{22,1}\Big{]}\end{bmatrix}  \tag{40}\]  
\[\text{with}\quad[\mathbf{K}_{11,2}]=[\mathbf{K}_{11}]-[\mathbf{K}_{12}]\Big{[} \mathbf{K}^{-1}_{22}\Big{]}[\mathbf{K}_{21}]\quad\text{and}\quad[\mathbf{K} _{11}]^{-1}=[\mathbf{K}_{11,2}] \tag{41}\]  
\([\mathbf{K}_{11}]\), \([\mathbf{K}_{22}]\), \([\mathbf{K}_{11,2}]\), and \([\mathbf{K}_{22,1}]\) are non-singular matrices [21]. Based on Equations (38), (40) and (41) the following Equation can be obtained:  
\[\Big{[}\mathbf{K}^{2}\Big{]}\Big{[}\mathbf{K}^{1}\Big{]}^{-1}=-[\mathbf{K}_{2 2}]^{-1}[\mathbf{K}_{21}] \tag{42}\]  
The maximum likelihood estimator of the SOMRS model in the population is partitioned to be  
\[\overset{\wedge}{\boldsymbol{\Theta}}= \begin{bmatrix}\overset{\wedge}{ \boldsymbol{\pi}}^{\prime}&\overset{\wedge}{\boldsymbol{\Theta}}_{1}\end{bmatrix} ^ {\prime} \tag{43}\]  
with \(\overset{\wedge}{\boldsymbol{\Theta}}_{10}= \begin{bmatrix}\hat{\psi}_{1}& \ldots&\hat{\psi}_{n}&\overset{\wedge}{\boldsymbol{\beta}}_{0}\end{bmatrix} ^ {\prime}\). If \(\left(\overset{\wedge}{\boldsymbol{\Theta}}-\boldsymbol{\Theta}\right)\overset {d}{\rightarrow}N\bigg{(}\boldsymbol{0},\bigg{[}\mathbf{K}\bigg{(}\overset{ \wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]}^{-1}\bigg{)},k\rightarrow\infty\) is known [22], and pays attention to the Equation (43), then  
\[ \begin{bmatrix}\overset{\wedge}{\boldsymbol{\beta}}^{*}-\overset{\wedge}{ \boldsymbol{\beta}}^{*}\\ \overset{\wedge}{\boldsymbol{\Theta}}_{1}-\boldsymbol{\Theta}_{1}\end{bmatrix}  \overset{d}{\rightarrow}N\bigg{(} \begin{bmatrix}\boldsymbol{0}_{(kq\times 1)} \\ \boldsymbol{0}_{(kq\times 1)}\end{bmatrix} ,\bigg{[}\mathbf{K}\bigg{(}\overset{ \wedge}{\boldsymbol{\Theta}}\bigg{)}\bigg{]}^{-1}\bigg{)},k\rightarrow\infty \tag{45}\]  
Based on the theory of multivariate normal distribution, the conditional distribution of \(\mathbf{y}\) if given \(\mathbf{x}\) is multivariate normal, i.e.,  
\[\mathbf{y}|\mathbf{x}\sim\mathbf{N}(E(\mathbf{y}|\mathbf{x}),\text{cov}( \mathbf{y}|\mathbf{x})) \tag{46}\]  
With \(E(\mathbf{y}|\mathbf{x})=\boldsymbol{\mu}_{\mathbf{y}}+\Sigma_{\mathbf{yx}} \Sigma_{\mathbf{xx}}^{-1}(\mathbf{x}-\boldsymbol{\mu}_{x})\) and \(\text{cov}(\mathbf{y}|\mathbf{x})=\Sigma_{\mathbf{yy}}-\Sigma_{\mathbf{yx}} \Sigma_{\mathbf{xx}}^{-1}(\mathbf{x}-\boldsymbol{\mu}_{x})\).