## 3 Proposed Method
### IG Model Considering Random Effects and Detection Errors
Let \(\mathbf{E}_{ij}=\big{(}E_{ij,0},E_{ij,1},\ldots,E_{ij,m_{i}}\big{)}^{T}\), \(\mathbf{E}_{ij}\) follows the multivariate normal distribution with mean 0 and variance \(\sigma_{\varepsilon}^{2}\mathbf{I}_{m_{i}+1}\), and \(\mathbf{I}_{m_{i}+1}\) is the \(m_{i+1}\)-dimensional identity matrix. Let \(\Delta E_{ij}=\mathbf{J}\mathbf{E}_{ij}\), and the matrix \(\mathbf{J}\) is as follows:  
\[\mathbf{J}= \begin{pmatrix}-1&1&0&\ldots&0\\ 0&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&-1&1&0\\ 0&\ldots&0&-1&1\end{pmatrix}  \tag{16}\]  
It is easy to see through a linear transformation of \(\Delta E_{ij}\) that \(\Delta E_{ij}\) follows the \(m_{i}\)-dimensional normal distribution with mean 0 and variance \(\sum_{\Delta E_{ij}}\). Then, the formula for \(\sum_{\Delta E_{ij}}\) is as follows:  
\[\sum_{\Delta E_{ij}}=\sigma_{\varepsilon}^{2}\mathbf{I}_{m_{i}+1}\mathbf{J}^{T}=\sigma_ {\varepsilon}^{2}\mathbf{J}^{T}=2\sigma_{\varepsilon}^{2} \begin{pmatrix}1&-1/2& \ldots&0\\ -1/2&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&-1/2\\ 0&\ldots&-1/2&1\end{pmatrix}  \tag{17}\]Then, the PDF of \(\sum_{\Delta E_{ij}}\) is as follows:  
\[f_{\Delta E_{ij}}(\Delta e_{ij})=\frac{1}{(2\pi)^{1/2}\Big{|}\Sigma_{\Delta E_{ij }}\Big{|}}\exp\left(-\frac{1}{2}\Delta e_{ij}\sum_{\Delta E_{ij}}^{-1}\Delta e _{ij}^{T}\right) \tag{18}\]  
where \(\Big{|}\Sigma_{\Delta E_{ij}}\Big{|}\) is the determinant of the variance matrix \(\Sigma_{\Delta E_{ij}}\). Combining Formulas (12) and (18), the likelihood function of the parameter \(\theta\) can be obtained by using the convolution formula:  
\[ \begin{split} L(\theta)=\prod\limits_{j=1}^{n}\prod\limits_{i=1} ^{k}\prod\limits_{l=1}^{m_{i}}f\left(\Delta y_{ij,l};\theta\right)=\prod \limits_{j=1}^{n}\prod\limits_{i=1}^{k}\prod\limits_{l=1}^{m_{i}}& \int_{-\infty}^{\Delta y_{ij,l}}\frac{\Gamma(\delta+1/2)}{ \Gamma(\delta)}\gamma^{\delta}\sqrt{\frac{\Delta t_{ij,l}^{2}}{2\pi\left( \Delta y_{ij,l}-\Delta e_{ij,l}\right)^{3}}}\\ &\times\left\{\gamma+\frac{\left[\left(\Delta y_{ij,l}-\Delta e _{ij,l}\right)-h(x_{i})\Delta t_{ij,l}\right]^{2}}{2h(x_{i})^{2}\left(\Delta y _{ij,l}-\Delta e_{ij,l}\right)}\right\}^{-\delta-1/2}\\ &\times\frac{1}{\left(2\pi\right)^{1/2}\Big{|}\Sigma_{\Delta E_{ ij,l}}\Big{|}}\exp\left(-\frac{1}{2}\Delta e_{ij,l}\Sigma_{\Delta E_{ij,l}}^{-1} \Delta e_{ij,l}\right)d\Delta e_{ij,l}\end{split}  \tag{19}\]  
where \(\Delta e_{ij,l}<\Delta y_{ij,l}\), and the corresponding logarithmic likelihood function is as follows:  
\[ \begin{split}\ln L(\theta)=\sum\limits_{j=1}^{n}\sum\limits_{i=1 }^{k}\sum\limits_{l=1}^{m_{i}}\ln&\int_{-\infty}^{\Delta y_{ij, j}}\frac{\Gamma(\delta+1/2)}{\Gamma(\delta)}\gamma^{\delta}\sqrt{\frac{ \Delta t_{ij,l}^{2}}{2\pi\left(\Delta y_{ij,l}-\Delta e_{ij,l}\right)^{3}}}\\ &\times\left\{\gamma+\frac{\left[\left(\Delta y_{ij,l}-\Delta e _{ij,l}\right)-h(x_{i})\Delta t_{ij,l}\right]^{2}}{2h(x_{i})^{2}\left(\Delta y _{ij,l}-\Delta e_{ij,l}\right)}\right\}^{-\delta-1/2}\\ &\times\frac{1}{\left(2\pi\right)^{1/2}\Big{|}\Sigma_{\Delta E_{ ij,l}}\Big{|}}\exp\left(-\frac{1}{2}\Delta e_{ij,l}\Sigma_{\Delta E_{ij,l}}^{-1} \Delta e_{ij,l}\right)d\Delta e_{ij,l}\end{split}  \tag{20}\]  
Therefore, the unknown parameters of the IG model considering random effects and detection errors are \(\theta=\left(\alpha_{0},\alpha_{1},\delta,\gamma,\sigma_{\epsilon}^{2}\right)^ {\prime}\).