# Model-Robust Estimation of Multiple-Group Structural Equation Models
## 2 Model-Robust Estimation in Structural Equation Modeling
### Robust Moment Estimation Using Robust Loss Functions
#### 2.2.2 Standard Error Estimation  
In this subsection, standard error formulas for RME are derived. Note that the parameter estimate \(\mathbf{\theta}\) is a nonlinear function of the vector of sufficient statistics \(\mathbf{\zeta}\) that contains group-wise mean vectors \(\mathbf{m}_{g}\) and nonduplicated elements of group-wise covariance matrices \(\mathbf{s}_{g}~{}=~{}\text{vech}(\mathbf{S}_{g})\). The covariance matrix \(\mathbf{V}_{\mathbf{\zeta}}~{}=~{}\text{Var}(\mathbf{\zeta})\) is derived under multivariate normality of observations \(\mathbf{X}\). The vectors \(\mathbf{m}_{g}\) and \(\mathbf{s}_{g}\) are uncorrelated within group \(g\) and across groups \(g=1,\ldots,G\). Hence, \(\mathbf{V}_{\mathbf{\zeta}}\) can be written as a block-diagonal matrix of covariance matrices \(\text{Var}(\mathbf{m}_{g})\) and \(\text{Var}(\mathbf{s}_{g})\). If \(N_{g}\) is the sample size of group \(g\), the covariance matrix of the mean vector is estimated by  
\[\text{Var}(\mathbf{m}_{g})~{}=~{}N_{g}^{-1}\mathbf{S}_{g}. \tag{15}\]  
Let \(\text{vec}(\mathbf{S})\) be the vectorized matrix of the covariance matrix \(\mathbf{S}\). Then, there exists a transition matrix \(\mathbf{K}\) that contains appropriate entries \(0\), \(0.5\), and \(1\) such that (see [3])  
\[\mathbf{s}_{g}~{}=~{}\text{vech}(\mathbf{S}_{g})~{}=~{}\text{Kvec}(\mathbf{S}_{g}). \tag{16}\]  
If \(\otimes\) denotes the Kronecker product, the covariance matrix of \(\mathbf{s}_{g}\) is estimated by (see [3])  
\[\text{Var}(\mathbf{s}_{g})~{}=~{}N_{g}^{-1}\mathbf{K}(\mathbf{S}_{g}\otimes\mathbf{S}_{g})\mathbf{ K}^{\top}. \tag{17}\]  
Hence, \(\mathbf{V}_{\mathbf{\zeta}}\) is given as a block-diagonal matrix with block-wise matrix entries defined by (15) and (17). RME can be viewed as obtaining the parameter estimate \(\hat{\mathbf{\theta}}\) by minimizing a (possibly robust) discrepancy and (approximated) differentiable function \(F(\mathbf{\theta},\mathbf{\zeta})\). Denote by \(F_{\mathbf{\theta}}~{}=~{}(\partial F)/(\partial\mathbf{\theta})\) the vector of partial derivatives with respect to \(\mathbf{\theta}\). The parameter estimate is given as the root of the nonlinear equation  
\[F_{\theta}(\hat{\mathbf{\theta}},\mathbf{\zeta})~{}=~{}\mathbf{0}~{}. \tag{18}\]  
Assume that the population sufficient statistics are denoted by \(\mathbf{\zeta}_{0}\) and there exists a pseudo-true parameter \(\mathbf{\theta}_{0}\) such that \(F_{\mathbf{\theta}}(\mathbf{\theta}_{0},\mathbf{\zeta}_{0})~{}=~{}\mathbf{0}\). Note that the parameter \(\mathbf{\zeta}_{0}\) does not refer to a data-generating parameter but is defined by choosing a particular discrepancy function \(F\). Different pseudo-true parameters will be obtained for different choices of discrepancy functions in misspecified SEMs. We now derive the covariance matrix of \(\hat{\mathbf{\theta}}\) by utilizing a [PERSON] expansion of \(F_{\mathbf{\theta}}\). Denote by \(F_{\mathbf{\theta}\theta}\) and \(F_{\mathbf{\theta}\zeta}\) the matrices of second-order partial derivatives of \(F_{\mathbf{\theta}}\) with respect to \(\mathbf{\theta}\) and \(\mathbf{\zeta}\), respectively.