## 3 Modeling and Parameter Estimation in CSADT
### Parameter Estimation  
Let \(\Lambda(t)=t^{\theta}\), the unknown parameters in the model are \(\Theta=\Big{\{}\mu_{\eta},\sigma_{\eta}^{2},\kappa,b,\theta,\sigma_{\rm e}^{2} \Big{\}}\), where \(\mu_{\eta}\) and \(\sigma_{\eta}^{2}\) are random coefficients, which are applied to characterize the individual degradation characteristics of the product concerned with lifetime prediction. Further, others are fixed coefficient models which representing the common degradation characteristics between similar products. They are dealt with through the MLE method. Assuming that \(N\) test samples are carried out a CSADT under the following conditions:  
(i) \(S_{0}<S_{1}<\cdots<S_{k}<\cdots S_{d}\) and \(d\) is the total stress levels, where \(S_{0}\) is the normal stress. (ii) There are \(n_{k}\) samples at a stress \(S_{k}\) for a degradation test, where \(\sum\limits_{k=1}^{d}n_{k}=\)\(N\), \(1\leq k\leq d\). (iii) The inspections are taken \(m\) times for each stress level and each sample is monitored at time \(t_{1},t_{2},\cdots t_{m}\). The observation interval is \(\Delta t=t_{j}-t_{j-1}\), \(j=1,2,\cdots,m\), then \(t_{m}=m\Delta t\). (iv) For \(1\leq i\leq n_{k}\), \(1\leq k\leq d\), \(1\leq j\leq m\), let \(\gamma_{i}\big{(}t_{j}|S_{k}\big{)}\) express the sample degradation process of \(ith\) sample at time \(t_{j}\) under \(S_{k}\). \[Y_{ijk}=Y_{i}\big{(}t_{j}|S_{k}\big{)}=\lambda_{k}\Lambda\big{(}t_{j}\big{)}+ \sigma_{\rm B}B\big{(}\Lambda\big{(}t_{j}\big{)}\big{)}+\sigma_{\rm e}\varepsilon \tag{20}\]  
For simplicity, let \(\Lambda=(\Lambda(t_{1}),\Lambda(t_{2}),\cdots,\Lambda(t_{m}))^{\prime},\mathbf{ Y}_{k}=(Y_{i}(t_{1}|S_{k}),Y_{i}(t_{2}|S_{k}),\cdots,Y_{i}(t_{m}|S_{k}))^{\prime}\) and \(\mathbf{Y}=\big{(}\mathbf{Y}_{11}{}^{\prime},\mathbf{Y}_{21}{}^{\prime},\cdots, \mathbf{Y}_{1n_{1}}{}^{\prime},\cdots,\mathbf{Y}_{1d}{}^{\prime},\mathbf{Y}_{ 2d}{}^{\prime},\cdots,\mathbf{Y}_{nd}{}^{\prime}\big{)}^{\prime}\). According to property described in Section 2, \(\mathbf{Y}_{ik}\) follows a multivariate normal distribution with mean \(\mu_{\eta}\exp(-b/S_{k})\Lambda\) and variance \(\Sigma_{k}=\sigma_{\eta}^{2}\exp(-2b/S_{k})\Lambda\Lambda^{{}^{\prime}}+ \mathbf{\Omega}\), where \(\mathbf{\Omega}=\kappa\exp(-b/S_{k})\mathbf{Q}+\sigma_{\rm e}^{2}\mathbf{I}_{m}\), \(\mathbf{I}_{m}\) is an identified matrix of order \(m\), and  
\[\mathbf{Q}=\left[ \begin{array}{cccc}\Lambda(t_{1})&\Lambda(t_{1})&\cdots& \Lambda(t_{1})\\ \Lambda(t_{1})&\Lambda(t_{2})&\cdots&\Lambda(t_{2})\\ \vdots&\vdots&\ddots&\vdots\\ \Lambda(t_{1})&\Lambda(t_{2})&\cdots&\Lambda(t_{m})\end{array} \right] \tag{21}\]  
Then the joint PDF for the observed vector \(\mathbf{Y}_{ik}\) is derived as\[f_{\mathbf{Y}_{ik}}(\mathbf{Y}_{ik})=(2\pi)^{-\frac{m_{k}}{2}}|\Sigma_{k}|^{-\frac {1}{2}}\cdot\exp\left(-\frac{\left(\mathbf{Y}_{ik}-\mu_{q}\exp(-b/S_{k})\Lambda \right)^{\prime}\cdot\Sigma_{k}^{-1}\cdot\left(\mathbf{Y}_{ik}-\mu_{q}\exp(-b/S _{k})\Lambda\right)}{2}\right) \tag{22}\]  
Thus, the log-likelihood function unknown parameters \(\Theta=\left\{\mu_{q},\sigma_{\eta}^{2},\kappa,b,\theta,\sigma_{\varepsilon}^ {2}\right\}\) is  
\[ \begin{split}\ln L(\Theta|\mathbf{Y})&=-\frac{Nm}{ 2}\ln(2\pi)-\frac{N}{2}\ln|\Sigma_{k}|\\ &-\frac{1}{2}\sum\limits_{k=1}^{d}\sum\limits_{i=1}^{m_{k}}\left( \mathbf{Y}_{ik}-\mu_{q}\exp(-b/S_{k})\Lambda\right)^{\prime}\cdot\Sigma_{k}^{ -1}\cdot\left(\mathbf{Y}_{ik}-\mu_{q}\exp(-b/S_{k})\Lambda\right)\end{split}  \tag{23}\]  
where \(|\Sigma_{k}|=|\mathbf{\Omega}|\left(1+\sigma_{\eta}^{2}\exp(-2b/S_{k})\Lambda ^{\prime}\mathbf{\Omega}^{-1}\Lambda\right)\), and \(\Sigma_{k}^{-1}=\mathbf{\Omega}^{-1}-\frac{\sigma_{\eta}^{2}\exp(-2b/S_{k})}{ 1+\sigma_{\eta}^{2}\exp(-2b/S_{k})\Lambda^{\prime}\mathbf{\Omega}^{-1}\Lambda}\). For simplicity, let \(\widetilde{\kappa}=\kappa/\sigma_{\eta}^{2},\widetilde{\sigma_{\varepsilon}^ {2}}=\sigma_{\varepsilon}^{2}/\sigma_{\eta}^{2}\), and \(\widetilde{\Sigma}_{k}=\Sigma_{k}/\sigma_{\eta}^{2}\).