Elliptical linear mixed models with a covariate subject to measurement error

Abstract

In this paper we extend linear mixed models with elliptical errors by adding a covariate subject to measurement error in the linear predictor. The former class is defined appropriately so that the joint marginal distribution of the response and the observed covariate subject to measurement error is also elliptical. Thus, numerical integration methods are not required to obtain the marginal model and the mean and the variance–covariance structures of the hierarchical model are preserved. A kurtosis flexibility is allowed for each joint marginal distribution and since the conditional distributions are also elliptical, the predictions of the random effects as well as of the covariate subject to measurement error may be performed in a similar way of the normal case. A reweighed iterative process based on the maximum likelihood method is derived for obtaining the parameter estimates, which appear to be robust against outlying observations in the sense of the Mahalanobis distance. In order to assess the sample properties of the maximum likelihood parameter estimates as well as their asymptotic standard errors, a simulation study is performed under different parameter settings and error distributions. Goodness-of-fit measures based on the Mahalanobis distance are presented and normal curvatures of local influence are derived under three usual perturbation schemes, which are selected appropriately. Finally, an illustrative example previously analyzed under normal error models is reanalyzed by considering heavy-tailed error models. The diagnostic procedures are applied for comparing the fitted models.

Appendices

Appendix A: Derivation of the Fisher information matrix

Consider the Mahalanobis distance expressed in the following form:

$$\begin{aligned} \delta _i= & {} ({\varvec{W}}_i-{\varvec{\mu }}_{iW})^{\top } {\varvec{V}}^{-\frac{1}{2}}_i{\varvec{V}}^{-\frac{1}{2}}_i({\varvec{W}}_i-{\varvec{\mu }}_{iW})\\= & {} {\varvec{r}}_i^{\top } {\varvec{V}}^{-\frac{1}{2}}_i{\varvec{V}}^{-\frac{1}{2}}_i{\varvec{r}}_i\\= & {} {\varvec{z}}_i^{\top } {\varvec{z}}_i\\= & {} \Vert {\varvec{z}}_i\Vert ^2, \end{aligned}$$

where $\Vert {\varvec{z}}_i\Vert$ denotes the norm of ${\varvec{z}}_i={\varvec{\Sigma }}^{-\frac{1}{2}}_i{\varvec{r}}_i$ and ${\varvec{z}}_i \mathop {\sim }\limits ^{\mathrm {ind}}\mathrm{El}_{m_i}(\mathbf{0}, {\varvec{I}}_{m_i})$, $i=1, \ldots , n$.

From Fang et al. (1990) one has

$$\begin{aligned} \mathrm{E}\left\{ W_g({\varvec{\delta }}_i)\Vert {\varvec{z}}_i\Vert ^2\right\}= & {} -\frac{m_i}{2},\\ \mathrm{E}\left\{ W^2_g({\varvec{\delta }}_i)\Vert {\varvec{z}}_i\Vert ^2\right\}= & {} d_{g_i} \ \ \mathrm{and} \\ \mathrm{E}\left\{ W^2_g({\varvec{\delta }}_i)\Vert {\varvec{z}}_i\Vert ^4\right\}= & {} f_{g_i}. \end{aligned}$$

In particular, for the normal distribution $d_{g_i}=\frac{m_i}{2}$ and $f_{g_i}=m_i(m_i+1)$, whereas for the Student-t distribution with $\nu$ degrees of freedom these quantities become $d_{g_i}=\frac{m_i}{2}\frac{\nu +2m_i}{\nu +2m_i+2}$ and $f_{g_i}=m_i(m_i+1)\frac{\nu +2m_i}{\nu +2m_i+2}$. Expression for $d_{g_i}$ and $f_{g_i}$ for other elliptical distributions may be found, for instance, in Osorio et al. (2007).

1.1 Fisher information for $\gamma$

The Fisher information for the parameter $\gamma$ is defined by

$$\begin{aligned} F_{\gamma \gamma }({\varvec{\theta }})= \mathrm{E}\left\{ U^\gamma ({\varvec{\theta }}) U^\gamma ({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }}) U^\gamma _i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }})U^\gamma _i({\varvec{\theta }})\right\}= & {} \frac{{\varvec{a}}_i^2}{4}\left( \frac{f_{g_i}}{m_i(m_i+1)}-1\right) + \frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }{\varvec{V}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) \\&+ \, 2d_{g_i}\left( \begin{array}{c} \mu \mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^{\top }{\varvec{V}}^{-1}_i\left( \begin{array}{c} \mu \mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) , \quad i=1,\ldots ,n, \end{aligned}$$

with $\mathbf{0}$ being an $m_i \times 1$ vector of zeros.

1.2 Fisher information submatrix for ${\varvec{\beta }}$

The Fisher information submatrix for the parameter vector ${\varvec{\beta }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\beta \beta }({\varvec{\theta }})=\mathrm{E}\left\{ {\varvec{U}}^{\beta }({\varvec{\theta }}) {\varvec{U}}^{\beta }{({\varvec{\theta }})^{\top }}\right\} =\sum _{i=1}^n \mathrm{E}\left\{ {\varvec{U}}^{\beta }_i({\varvec{\theta }}) {\varvec{U}}^{\beta }_i({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ {\varvec{U}}^{\beta }_i({\varvec{\theta }}) {\varvec{U}}^{\beta }_i({\varvec{\theta }})^{\top }\right\}= & {} \frac{2d_{g_i}}{m_i}\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^{\top }{\varvec{V}}^{-1}_i\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) , \end{aligned}$$

and ${\varvec{X}}_i$ is an $m_i \times p$ matrix with the explanatory variable values for the ith observation, $i=1,\ldots ,n$.

1.3 Fisher information for $\mu$

The Fisher information for the parameter $\mu$ is defined by

$$\begin{aligned} F_{\mu \mu }({\varvec{\theta }})= \mathrm{E}\left\{ U^\mu ({\varvec{\theta }}) U^\mu ({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^\mu _i({\varvec{\theta }}) U^\mu _i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^\mu _i({\varvec{\theta }}) U^{\mu }_i({\varvec{\theta }})\right\}= & {} \frac{2d_{g_i}}{m_i}\left( \begin{array}{c} \gamma \mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top {\varvec{V}}^{-1}_i\left( \begin{array}{c} \gamma \mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) , \ i=1,\ldots ,n. \end{aligned}$$

1.4 Fisher information for $\sigma ^2_u$

The Fisher information for the parameter $\sigma ^2_u$ is defined by

$$\begin{aligned} F_{\sigma ^2_u \sigma ^2_u}({\varvec{\theta }})= \mathrm{E}\left\{ U^{\sigma ^2_u}({\varvec{\theta }}) U^{\sigma ^2_u}({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }}) U^{\sigma ^2_u}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }})U^{\sigma ^2_u}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits ^2\left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.5 Fisher information for $\sigma ^2$

The Fisher information for the parameter $\sigma ^2$ is defined by

$$\begin{aligned} F_{\sigma ^2 \sigma ^2}({\varvec{\theta }})= \mathrm{E}\left\{ U^{\sigma ^2}({\varvec{\theta }}) U^{\sigma ^2}({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^{\sigma ^2}_i({\varvec{\theta }}) U^{\sigma ^2}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\sigma ^2}_i({\varvec{\theta }})U^{\sigma ^2}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits ^2\left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.6 Fisher information submatrix for ${\varvec{\tau }}$

The Fisher information submatrix for the parameter vector ${\varvec{\tau }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\tau \tau }({\varvec{\theta }})=\mathrm{E}\left\{ {\varvec{U}}^{\tau }({\varvec{\theta }}) {\varvec{U}}^{\tau }({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

with the (j, l)th element being expressed as

$$\begin{aligned} \mathrm{E}\left\{ U^{\tau _j}({\varvec{\theta }})U^{\tau _l}({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^{\tau _j}_i({\varvec{\theta }})U^{\tau _l}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\tau _j}_i({\varvec{\theta }})U^{\tau _l}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _l}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _l}\right) , \ i=1,\ldots ,n. \end{aligned}$$

1.7 Fisher information submatrix for $\gamma$ and ${\varvec{\beta }}$

The Fisher information submatrix for the parameter $\gamma$ and the vector ${\varvec{\beta }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\gamma \beta }({\varvec{\theta }})=\mathrm{E}\left\{ U^{\gamma }({\varvec{\theta }}) {\varvec{U}}^{\beta }({\varvec{\theta }})^{\top }\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }}){\varvec{U}}^{\beta }_i({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }}){\varvec{U}}^{\beta }_i({\varvec{\theta }})^{\top }\right\}= & {} \frac{2d_{g_i}}{m_i}\left( \begin{array}{c} \mu \mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^{\top }{\varvec{V}}^{-1}_i\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) , \ i=1,\ldots ,n. \end{aligned}$$

1.8 Fisher information for $\gamma$ and $\mu$

The Fisher information for the parameters $\gamma$ and $\mu$ is defined by

$$\begin{aligned} F_{\gamma \mu }({\varvec{\theta }})= \mathrm{E}\left\{ U^\gamma ({\varvec{\theta }}) U^\mu ({\varvec{\theta }})\right\} = \sum _{i=1}^n \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }}) U^\mu _i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }})U^{\mu }_i({\varvec{\theta }})\right\}= & {} \frac{2d_{g_i}}{m_i}\left( \begin{array}{c} \mu \mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^{\top }{\varvec{V}}^{-1}_i\left( \begin{array}{c} \gamma \mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.1 Fisher information for $\gamma$ and $\sigma ^2_u$

The Fisher information for the parameters $\gamma$ and $\sigma ^2_u$ is defined by

$$\begin{aligned} F_{\gamma \sigma ^2_u}({\varvec{\theta }})= \sum _{i=1}^n \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }})U^{\sigma ^2_u}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }})U^{\sigma ^2_u}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma } {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.2 Fisher information for $\gamma$ and $\sigma ^2$

The Fisher information for the parameters $\gamma$ and $\sigma ^2$ is defined by

$$\begin{aligned} F_{\gamma \sigma ^2}({\varvec{\theta }})= \sum _{i=1}^n \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }})U^{\sigma ^2}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }})U^{\sigma ^2}_i({\varvec{\theta }}) \right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.3 Fisher information submatrix for $\gamma$ and ${\varvec{\tau }}$

The Fisher information submatrix for the parameters $\gamma$ and ${\varvec{\tau }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\gamma \tau }({\varvec{\theta }})= \sum _{i=1}^n \mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }}){\varvec{U}}^\tau _i({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

where $\mathrm{E}\left\{ U^\gamma _i({\varvec{\theta }}){\varvec{U}}^\tau _i({\varvec{\theta }})^{\top }\right\}$ is an $1 \times (q+s)$ vector whose jth element is given by

$$\begin{aligned} \mathrm{E}\left\{ U^{\gamma }_i({\varvec{\theta }})U^{\tau _j}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.4 Fisher information submatrix for ${\varvec{\beta }}$ and $\mu$

The Fisher information submatrix for the parameters ${\varvec{\beta }}$ and $\mu$ is defined by

$$\begin{aligned} {\varvec{F}}_{\beta \mu }({\varvec{\theta }})= \sum _{i=1}^n\mathrm{E}\left\{ {\varvec{U}}^\beta _i({\varvec{\theta }}) U^{\mu }_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ {\varvec{U}}^{\beta }_i({\varvec{\theta }})U^{\mu }_i({\varvec{\theta }})\right\}= & {} \frac{2d_{g_i}}{m_i}\left( \begin{array}{c} {\varvec{X}}_i\\ \mathbf{0} \\ \end{array}\right) ^{\top }{\varvec{V}}^{-1}_i\left( \begin{array}{c} \gamma \mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.5 Fisher information for $\sigma ^2_u$ and $\sigma ^2$

The Fisher information for the parameters $\sigma ^2_u$ and $\sigma ^2$ is defined by

$$\begin{aligned} F_{\sigma ^2_u\sigma ^2}({\varvec{\theta }})=\sum _{i=1}^n \mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }}) U^{\sigma ^2}_i({\varvec{\theta }})\right\} , \end{aligned}$$

where

$$\begin{aligned} \mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }})U^{\sigma ^2}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u} {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.8.6 Fisher information submatrix for $\sigma ^2_u$ and ${\varvec{\tau }}$

The Fisher information submatrix for the parameter $\sigma ^2_u$ and the vector ${\varvec{\tau }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\sigma ^2_u\tau }({\varvec{\theta }})=\sum _{i=1}^n\mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }}) {\varvec{U}}^{\tau }_i({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

where $\mathrm{E}\{U^{\sigma ^2_u}_i({\varvec{\theta }}){\varvec{U}}^\tau _i({\varvec{\theta }})^{\top }\}$ is an $1 \times (q+s)$ vector whose jth element is given by

$$\begin{aligned} \mathrm{E}\left\{ U^{\sigma ^2_u}_i({\varvec{\theta }})U^{\tau _j}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

1.9 Fisher information submatrix for $\sigma ^2$ and ${\varvec{\tau }}$

The Fisher information submatrix for the parameter $\sigma ^2_u$ and the vector ${\varvec{\tau }}$ is defined by

$$\begin{aligned} {\varvec{F}}_{\sigma ^2\tau }({\varvec{\theta }})=\sum _{i=1}^n\mathrm{E}\left\{ U^{\sigma ^2}_i({\varvec{\theta }}) {\varvec{U}}^{\tau }_i({\varvec{\theta }})^{\top }\right\} , \end{aligned}$$

where $\mathrm{E}\left\{ U^{\sigma ^2}_i({\varvec{\theta }}){\varvec{U}}^\tau _i({\varvec{\theta }})^{\top }\right\}$ is an $1 \times (q+s)$ vector whose jth element is given by

$$\begin{aligned} \mathrm{E}\left\{ U^{\sigma ^2}_i({\varvec{\theta }})U^{\tau _j}_i({\varvec{\theta }})\right\}= & {} \frac{1}{4}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2} \right) \mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) \left\{ \frac{f_{g_i}}{m_i(m_i+1)}-1\right\} \\&+\,\frac{f_{g_i}}{2m_i(m_i+1)}\mathop {\mathrm{tr}}\nolimits \left( {\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}{\varvec{V}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\right) , \quad i=1,\ldots ,n. \end{aligned}$$

Applying similar calculation we may show the orthogonality between ${\varvec{\beta }}$ and $({\varvec{\tau }}^{\top }, \sigma ^2_u, \sigma ^2)^{\top }$ as well as between $\mu$ and $({\varvec{\tau }}^{\top }, \sigma ^2_u, \sigma ^2)^{\top }$.

Appendix B: Derivation of the Hessian matrix

The Hessian matrix for ${\varvec{\theta }}$ may be expressed as

$$\begin{aligned} \ddot{{\varvec{L}}}({\varvec{\theta }})=\sum _{i=1}^n\ddot{{\varvec{L}}}_i({\varvec{\theta }}), \end{aligned}$$

where

$$\begin{aligned} \ddot{{\varvec{L}}}_i({\varvec{\theta }})= & {} \frac{\partial ^2L_i({\varvec{\theta }})}{\partial {\varvec{\theta }}\partial {\varvec{\theta }}^\top }\\= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \ddot{L}_i^{\gamma \gamma } &{} \ddot{{\varvec{L}}}_i^{\gamma \beta } &{} \ddot{L}_i^{\gamma \mu } &{} \ddot{L}_i^{\gamma \sigma ^2_u} &{} \ddot{L}_i^{\gamma \sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\gamma \tau }\\ \ddot{{\varvec{L}}}_i^{\beta \gamma } &{} \ddot{{\varvec{L}}}_i^{\beta \beta } &{} \ddot{{\varvec{L}}}_i^{\beta \mu } &{} \ddot{{\varvec{L}}}_i^{\beta \sigma ^2_u} &{} \ddot{{\varvec{L}}}_i^{\beta \sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\beta \tau }\\ \ddot{L}_i^{\mu \gamma } &{} \ddot{{\varvec{L}}}_i^{\mu \beta } &{} \ddot{L}_i^{\mu \mu } &{} \ddot{L}_i^{\mu \sigma ^2_u} &{} \ddot{L}_i^{\mu \sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\mu \tau }\\ \ddot{L}_i^{\sigma ^2_u\gamma } &{} \ddot{{\varvec{L}}}_i^{\sigma ^2_u\beta } &{} \ddot{L}_i^{\sigma ^2_u\mu } &{} \ddot{L}_i^{\sigma ^2_u\sigma ^2_u} &{} \ddot{L}_i^{\sigma ^2_u\sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\sigma ^2_u\tau }\\ \ddot{L}_i^{\sigma ^2\gamma } &{} \ddot{{\varvec{L}}}_i^{\sigma ^2\beta } &{} \ddot{L}_i^{\sigma ^2\mu } &{} \ddot{L}_i^{\sigma ^2\sigma ^2_u} &{} \ddot{L}_i^{\sigma ^2\sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\sigma ^2\tau }\\ \ddot{{\varvec{L}}}_i^{\tau \gamma } &{} \ddot{{\varvec{L}}}_i^{\tau \beta } &{} \ddot{{\varvec{L}}}_i^{\tau \mu } &{} \ddot{{\varvec{L}}}_i^{\tau \sigma ^2_u} &{} \ddot{{\varvec{L}}}_i^{\tau \sigma ^2} &{} \ddot{{\varvec{L}}}_i^{\tau \tau } \end{array}\right) . \end{aligned}$$

Below we derive the submatrices above evaluated at $\hat{{\varvec{\theta }}}$. One has

$$\begin{aligned} \ddot{L}_i^{\gamma \gamma }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2L_i({\varvec{\theta }})}{\partial \gamma \partial \gamma }\right| _{\theta =\widehat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left\{ \widehat{{\varvec{V}}}^{-1}_i\frac{\partial {\varvec{V}}_i}{\partial \gamma }\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right\} -\frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \widehat{{\varvec{V}}}_i^{-1}\frac{\partial ^2{\varvec{V}}_i}{\partial \gamma ^2}\right) \\&+\ W'_g(\widehat{\delta }_i)\left\{ -2\widehat{{\varvec{r}}}_i^\top {\varvec{V}}^{-1}_i\left( \begin{array}{c} \widehat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\widehat{{\varvec{r}}}_i^\top \frac{\partial {\varvec{V}}_i^{-1}}{\partial \gamma }\widehat{{\varvec{r}}}_i\right\} ^2\\&+\ W_g(\widehat{\delta }_i)\left\{ 2\left( \begin{array}{c} \widehat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^\top \widehat{{\varvec{V}}}^{-1}_i\left( \begin{array}{c} \widehat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +4\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\widehat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \widehat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) \right\} \\&+\ W_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\left\{ 2\frac{\partial {\varvec{V}}_i}{\partial \gamma }\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }-\frac{\partial ^2{\varvec{V}}_i}{\partial \gamma ^2}\right\} \widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\beta \beta }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2\L _i({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial {\varvec{\beta }}^\top }\right| _{\theta =\widehat{\theta }}\\= & {} 2\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \widehat{{\varvec{V}}}_i^{-1}\left\{ 2W'_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) +W_g(\widehat{\delta }_i)\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) \right\} , \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\mu \mu }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \mu \partial \mu }\right| _{\theta =\widehat{\theta }}\\= & {} 2\left( \begin{array}{c} \widehat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \widehat{{\varvec{V}}}_i^{-1}\left\{ 2W'_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \widehat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) +W_g(\widehat{\delta }_i)\left( \begin{array}{c} \widehat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) \right\} , \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\sigma ^2_u\sigma ^2_u}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \sigma ^2_u\partial \sigma ^2_u}\right| _{\theta =\widehat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) +W'_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\\&+\ 2W_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\sigma ^2\sigma ^2}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \sigma ^2\partial \sigma ^2}\right| _{\theta =\widehat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) +W'_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\\&+\ 2W_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\tau \tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial {\varvec{\tau }}\partial {\varvec{\tau }}^\top }\right| _{\theta =\widehat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\right) +W'_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\\&+\ 2W_g(\widehat{\delta }_i)\widehat{{\varvec{r}}}_i^\top \widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\widehat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\gamma \beta }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \gamma \partial {\varvec{\beta }}^\top }\right| _{\theta =\hat{\theta }}\\= & {} \left\{ 2W'_g(\widehat{{\varvec{\delta }}}_i)\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\widehat{{\varvec{r}}}^\top _i + W_g(\widehat{{\varvec{\delta }}}_i)\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \right\} \widehat{{\varvec{V}}}_i^{-1}\left\{ 2\left( \begin{array}{c} \widehat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) \right. \\&\left. + \frac{\partial {\varvec{V}}_i}{\partial \gamma }\widehat{{\varvec{V}}}_i^{-1}\widehat{{\varvec{r}}}_i\right\} , \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\gamma \mu }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \gamma \partial \mu }\right| _{\theta =\hat{\theta }}\\= & {} 2W'_g(\hat{{\varvec{\delta }}}_i)\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\left\{ 2\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \\&+ 2W_g(\hat{{\varvec{\delta }}}_i)\left\{ \left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) \right. \\&\left. + \left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\gamma \sigma ^2_u}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \gamma \partial \sigma ^2_u}\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) -\frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial ^2{\varvec{V}}_i}{\partial \gamma \partial \sigma ^2_u}\right) \\&-W'_g(\hat{\delta }_i)\left\{ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \\&-W_g(\hat{\delta }_i)\left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) + \hat{{\varvec{r}}}_i^\top \hat{{\varvec{G}}}_i^{\sigma ^2_u}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

where $\hat{{\varvec{G}}}_i^{\sigma ^2_u}=\frac{\partial ^2{\varvec{V}}_i^{-1}}{\partial \gamma \partial \sigma ^2_u}=\hat{{\varvec{V}}}_i^{-1}\left( \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }-\frac{\partial ^2{\varvec{V}}_i}{\partial \gamma \partial \sigma ^2_u} +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) \hat{{\varvec{V}}}_i^{-1}$,

$$\begin{aligned} \ddot{L}_i^{\gamma \sigma ^2}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \gamma \partial \sigma ^2}\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) -W_g(\hat{\delta }_i)\left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) + \hat{{\varvec{r}}}_i^\top \hat{{\varvec{G}}}_i^{\sigma ^2}\hat{{\varvec{r}}}_i\right\} \\&-W'_g(\hat{\delta }_i)\left\{ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

where $\hat{{\varvec{G}}}_i^{\sigma ^2}=\frac{\partial ^2{\varvec{V}}_i^{-1}}{\partial \gamma \partial \sigma ^2}=\hat{{\varvec{V}}}_i^{-1}\left( \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma } +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) \hat{{\varvec{V}}}_i^{-1}$,

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\gamma \tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \gamma \partial {\varvec{\tau }}}\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) -W_g(\hat{\delta }_i)\left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\hat{{\varvec{V}}}_i^{-1}\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) + \hat{{\varvec{r}}}_i^\top \hat{{\varvec{G}}}_i^{\tau }\hat{{\varvec{r}}}_i\right\} \\&-\, W'_g(\hat{\delta }_i)\left\{ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \left\{ 2\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

where $\hat{{\varvec{G}}}_i^{\tau }=\frac{\partial ^2{\varvec{V}}_i^{-1}}{\partial \gamma \partial {\varvec{\tau }}}=\hat{{\varvec{V}}}_i^{-1}\left( \frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma } +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}}\right) \hat{{\varvec{V}}}_i^{-1}$,

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\beta \mu }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial \mu }\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i)\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) +2W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\beta \sigma ^2_u}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial \sigma ^2_u}\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\beta \sigma ^2}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial \sigma ^2}\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\beta \tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial {\varvec{\tau }}^\top }\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\mu \sigma ^2_u}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \mu \partial \sigma ^2_u}\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\mu \sigma ^2}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \mu \partial \sigma ^2}\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\mu \tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \mu \partial {\varvec{\tau }}^\top }\right| _{\theta =\hat{\theta }}\\= & {} 2\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\left\{ W_g(\hat{{\varvec{\delta }}}_i){\varvec{I}}_{2m_i}+W'_g(\hat{{\varvec{\delta }}}_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\right\} \frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{L}_i^{\sigma ^2_u\sigma ^2}(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \sigma ^2_u\partial \sigma ^2}\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) \\&+ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\left\{ W'_g(\hat{\delta }_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} + 2W_g(\hat{\delta }_i){\varvec{I}}_{2m_i}\right\} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\sigma ^2_u\tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \sigma ^2_u\partial {\varvec{\tau }}^\top }\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\right) \\&+ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\left\{ W'_g(\hat{\delta }_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} + 2W_g(\hat{\delta }_i){\varvec{I}}_{2m_i}\right\} \frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i \end{aligned}$$

and

$$\begin{aligned} \ddot{{\varvec{L}}}_i^{\sigma ^2\tau }(\hat{{\varvec{\theta }}})= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }})}{\partial \sigma ^2\partial {\varvec{\tau }}^\top }\right| _{\theta =\hat{\theta }}\\= & {} \frac{1}{2}\mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\right) \\&+\, \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\left\{ W'_g(\hat{\delta }_i)\hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} + 2W_g(\hat{\delta }_i){\varvec{I}}_{2m_i}\right\} \frac{\partial {\varvec{V}}_i}{\partial {\varvec{\tau }}^\top }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i. \end{aligned}$$

Appendix C: Derivation of ${\varvec{\Delta }}$ matrix

1.1 Case-weight perturbation

In particular for the normal case we obtain

$$\begin{aligned} {\varvec{\Delta }}_{\gamma }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \gamma \partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} -\frac{\sqrt{m_i}}{2}\left[ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\left\{ 2\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \right] , \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\beta } = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial {\varvec{\beta }}\partial \omega _i}\right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=\sqrt{m_i}\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\mu } = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \mu \partial \omega _i}\right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=\sqrt{m_i}\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\sigma ^2_u} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2_u\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\frac{\sqrt{m_i}}{2} \left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u} \right) -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\sigma ^2} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\frac{\sqrt{m_i}}{2} \left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2} \right) -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \end{aligned}$$

and

$$\begin{aligned} \left. {\varvec{\Delta }}_{\tau _j} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \tau _j\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\frac{\sqrt{m_i}}{2} \left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j} \right) -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

where $\hat{\delta }_i=\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\hat{{\varvec{r}}}_i$ and $\hat{{\varvec{r}}}_i={\varvec{W}}_i-\hat{{\varvec{\mu }}}_{iW}$, for $j=1, \ldots , q+s$.

For the Student-t case we obtain

$$\begin{aligned} {\varvec{\Delta }}_{\gamma }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \gamma \partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} -\frac{p_i}{2} \left[ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right) -v(\hat{\delta }_i)\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\left\{ 2\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) +\frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \right] , \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\beta } = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial {\varvec{\beta }}\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=p_i v(\hat{\delta }_i)\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\mu } = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \mu \partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=p_i v(\hat{\delta }_i)\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\sigma ^2} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2_u\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\frac{p_i}{2} \left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\right) - v(\hat{\delta }_i)\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

$$\begin{aligned} \left. {\varvec{\Delta }}_{\sigma ^2} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2 \partial \omega _i}\right| _{\theta =\widehat{\theta },\ \omega =\omega _0}= -\frac{p_i}{2}\left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\right) -v(\hat{\delta }_i)\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \end{aligned}$$

and

$$\begin{aligned} \left. {\varvec{\Delta }}_{\tau _j} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \tau _j\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\frac{p_i}{2} \left\{ \mathop {\mathrm{tr}}\nolimits \left( \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j} \right) -v(\hat{\delta }_i)\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} , \end{aligned}$$

where $p_i=\frac{2m_i+\nu }{2}\sqrt{\varPsi '\left( \frac{\nu }{2}\right) - \varPsi '\left( \frac{\nu +2m_i}{2}\right) }$, $\hat{\delta }_i=\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\hat{{\varvec{r}}}_i$ and $\hat{{\varvec{r}}}_i={\varvec{W}}_i-\hat{{\varvec{\mu }}}_{iW}$, fo $j=1, \ldots , q+s$.

1.2 Scale matrix perturbation

In general we obtain

$$\begin{aligned} {\varvec{\Delta }}_{\gamma }= & {} \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \gamma \partial \omega _i} \Bigg |_{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} \sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i)\right\} \left\{ 2\left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^\top -\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \gamma }\right\} \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i,\\ {\varvec{\Delta }}_{\beta }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial {\varvec{\beta }}\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-2\sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i)\right\} \left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i,\\ {\varvec{\Delta }}_{\mu }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \mu \partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-2\sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i)\right\} \left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i,\\ {\varvec{\Delta }}_{\sigma ^2_u}= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2_u\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i)\right\} \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i;\\ {\varvec{\Delta }}_{\sigma ^2}= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i) \right\} \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i \end{aligned}$$

and

$$\begin{aligned} \left. {\varvec{\Delta }}_{\tau _j} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \tau _j\partial \omega _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}=-\sqrt{f_{g_i}} \left\{ W'_g(\hat{\delta }_i)\hat{\delta }_i+W_g(\hat{\delta }_i) \right\} \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j}\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i, \end{aligned}$$

where $\hat{\delta }_i=\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\hat{{\varvec{r}}}_i$ and $\hat{{\varvec{r}}}_i={\varvec{W}}_i-\hat{{\varvec{\mu }}}_{iW}$, for $j=1, \ldots , q+s$.

1.3 Observed response perturbation

In general we obtain

$$\begin{aligned} {\varvec{\Delta }}_{\gamma }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \gamma \partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} -\frac{2}{\sqrt{2d_{g_i}/m_i}}\left[ W_g(\hat{\delta }_i)\left\{ \left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) ^\top \hat{{\varvec{V}}}_i^{-\frac{1}{2}}+ \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\left( \frac{\partial {\varvec{V}}^{\frac{1}{2}}_i}{\partial \gamma } - \frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\right) \hat{{\varvec{V}}}_i^{\frac{1}{2}}\right\} \right. \\&\left. +W'_g(\hat{\delta }_i)\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} \times \left\{ \left( \begin{array}{c} \hat{\mu }\mathbf{1}_{m_i} \\ \mathbf{0} \\ \end{array}\right) - \frac{\partial {\varvec{V}}_i}{\partial \gamma }\hat{{\varvec{V}}}_i^{-1}\hat{{\varvec{r}}}_i\right\} \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right] ,\\ {\varvec{\Delta }}_{\beta }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial {\varvec{\beta }}\partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} -\frac{2}{\sqrt{2d_{g_i}/m_i}}\left( \begin{array}{c} {\varvec{X}}_i \\ \mathbf{0} \\ \end{array}\right) ^\top \left\{ W_g(\hat{\delta }_i){\varvec{I}}_{2m_i}+2W'_g(\hat{\delta }_i) \hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \right\} \hat{{\varvec{V}}}_i^{\frac{1}{2}}\hat{{\varvec{V}}}_i^{-1},\\ {\varvec{\Delta }}_{\mu }= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \mu \partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} -\frac{2}{\sqrt{2d_{g_i}/m_i}}\left( \begin{array}{c} \hat{\gamma }\mathbf{1}_{m_i} \\ \mathbf{1}_{m_i} \\ \end{array}\right) ^\top \left\{ W_g(\hat{\delta }_i){\varvec{I}}_{2m_i}+2W'_g(\hat{\delta }_i) \hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \right\} \hat{{\varvec{V}}}_i^{\frac{1}{2}}\hat{{\varvec{V}}}_i^{-1},\\ {\varvec{\Delta }}_{\sigma ^2_u}= & {} \left. \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2_u\partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\= & {} \frac{2}{\sqrt{2d_{g_i}/m_i}}\left\{ W_g(\hat{\delta }_i)\left( \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1} \frac{\partial {\varvec{V}}_i^{\frac{1}{2}}}{\partial \sigma ^2_u} - \hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1} \frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right) \right. \\&\left. -W'_g(\hat{\delta }_i)\hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2_u}\hat{{\varvec{V}}}_i^{-1} \hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right\} , \end{aligned}$$

$$\begin{aligned}&\left. {\varvec{\Delta }}_{\sigma 2} - \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \sigma ^2\partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\&\quad =\frac{2}{\sqrt{2d_{g_i}/m_i}} \left\{ W_g(\hat{\delta }_i)\left( \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i^{1/2}}{\partial \sigma ^2} - \hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right) \right. \\&\qquad \left. -W'_g(\hat{\delta }_i) \hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \sigma ^2}\hat{{\varvec{V}}}_i^{-1} \hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right\} \end{aligned}$$

and

$$\begin{aligned}&\left. {\varvec{\Delta }}_{\tau _j} = \frac{\partial ^2 L_i({\varvec{\theta }}|{\varvec{\omega }})}{\partial \tau _j\partial {\varvec{\omega }}^\top _i} \right| _{\theta =\widehat{\theta },\ \omega =\omega _0}\\&\quad =\frac{2}{\sqrt{2d_{g_i}/m_i}} \left\{ W_g(\hat{\delta }_i)\left( \hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i^{\frac{1}{2}}}{\partial \tau _j} - \hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j} \hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right) \right. \\&\qquad \left. -W'_g(\hat{\delta }_i) \hat{{\varvec{r}}}^\top _i\hat{{\varvec{V}}}_i^{-1}\frac{\partial {\varvec{V}}_i}{\partial \tau _j} \hat{{\varvec{V}}}_i^{-1} \hat{{\varvec{r}}}_i\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}_i^{-\frac{1}{2}}\right\} , \end{aligned}$$

where $\hat{\delta }_i=\hat{{\varvec{r}}}_i^\top \hat{{\varvec{V}}}^{-1}_i\hat{{\varvec{r}}}_i$ and $\hat{{\varvec{r}}}_i={\varvec{W}}_i-\hat{{\varvec{\mu }}}_{iW}$, for $j=1, \ldots , q+s$.