## 3 Proposed Method
### IG Model Considering Random Effects and Detection Errors  
In this section, a new IG process model is proposed that takes into account random effects and detection errors. The specific details are as follows:  
Inspired by the literature [26], the Gamma distribution is introduced to characterize the randomness of and difference in the product degradation path to solve the problem posed by random effects on the product storage life assessment. It is assumed that \(\lambda\) follows the Gamma distribution, that is, \(\lambda\sim Gamma\big{(}\delta,\gamma^{-1}\big{)}\), \((\delta,\gamma>0)\). Its PDF can be expressed as follows:  
\[g(\lambda;\delta,\gamma)=\frac{\gamma^{\delta}\lambda^{\delta-1}}{\Gamma( \delta)}\exp(-\gamma\lambda)\,I_{(0,\infty)}(\lambda),\lambda>0 \tag{10}\]  
where \(\Gamma(\delta)\) is the Gamma function and \(\Gamma(\delta)=\int_{0}^{\infty}\lambda^{\delta-1}e^{-\lambda}d\lambda\), \(I_{(0,\infty)}(\lambda)=\left\{ \begin{array}{l}1,x\in(0,\infty)\\ 0,x\
otin(0,\infty)\end{array} \right..\) Combining Formulas (4) and (10), the PDF of \(Y(t)\) is as follows:  
\[f(y(t)|h(x),\delta,\gamma)=\frac{\Gamma(\delta+1/2)}{\Gamma(\delta)}\gamma^{ \delta}\sqrt{\frac{\Lambda^{2}(t)}{2\pi y(t)^{3}}}\times\left[\gamma+\frac{(y (t)-h(x)\Lambda(t))^{2}}{2h(x)^{2}y(t)}\right]^{-\delta-1/2} \tag{11}\]  
where the mean of \(Y(t)\) is \(\mu\Lambda(t)\), the variance is \(\gamma\mu^{3}\Lambda(t)/(\delta-1)\), and \(\mu=h(x),\delta>1\). The likelihood function for the IG process model considering random effects is as follows:\[L(\mathbf{\theta})=\prod_{j=1}^{n}\prod_{i=1}^{k}\prod_{l=1}^{m_{i}}f\Big{(}\Delta y_{ ij,l};\mathbf{\theta}\Big{)}=\prod_{j=1}^{n}\prod_{i=1}^{k}\prod_{l=1}^{m_{i}}\frac{ \Gamma(\delta+1/2)}{\Gamma(\delta)}\gamma^{\delta}\sqrt{\frac{\Delta t_{ij,l}^ {2}}{2\pi\Delta y_{ij,l}^{3}}}\times\left[\gamma+\frac{\Big{(}\Delta y_{ij,l}- h(x_{i})\Delta t_{ij,l}\Big{)}^{2}}{2h(x_{i})^{2}\Delta y_{ij,l}}\right]^{- \delta-1/2} \tag{12}\]  
where \(\mathbf{\theta}=\left(\alpha_{0},\alpha_{1},\delta,\gamma\right)^{\prime}\) is the unknown parameter. Therefore, its logarithmic likelihood function is as follows:  
\[ \begin{split}\ln L(\mathbf{\theta})=\sum_{j=1}^{n}\sum_{i=1}^{k} \sum_{l=1}^{m_{i}}&\left\{\ln\Gamma(\delta+m_{i}/2)-\ln\Gamma( \delta)\times\delta\ln\gamma+\Big{(}\ln\Delta t_{ij,l}-3/2\times\ln\Delta y_{ ij,l}\Big{)}\right.\\ &\left.-(\delta+m_{i}/2)\times\ln\left[\gamma+\frac{\Big{(}\Delta y _{ij,l}-e^{\hat{\alpha}_{0}+\hat{\alpha}_{1}x_{i}}\Delta t_{ij,l}\Big{)}^{2}} {2e^{2\hat{\alpha}_{0}+2\hat{\alpha}_{1}x_{i}}\Delta y_{ij,l}}\right]\right\} \end{split}  \tag{13}\]  
Considering the influence of uncertain factors such as detection equipment and environment, there are detection errors in the detection results. Therefore, this article uses the normal distribution to describe the detection errors of performance parameters. Assume \(Y_{ij,l}\) is the degradation detected value of the \(j\)-th sample at the time \(t_{l}\) under stress level \(S_{i}\), where \(i=1,2,\ldots,k;j=1,2,\ldots,n;l=1,2,\ldots,m_{i}\). After the ADT, the true value of the product degradation path is \(X_{ij,l}\), and then there is the following relationship between the detected value and the true value:  
\[Y_{ij,l}=X_{ij,l}+E_{ij,l} \tag{14}\]  
where \(E_{ij,l}\) is the detection error, and \(E_{ij,l}\) follows the normal distribution with mean 0 and standard deviation \(\sigma_{\varepsilon}\). Let \(\Delta y_{ij,l}=y_{ij,l}-y_{ij,l-1}\) be the increment of the two detection values, according to Formula (14), it is known that  
\[\Delta y_{ij,l}=\Delta x_{ij,l}+\Delta e_{ij,l} \tag{15}\]  
where \(\Delta e_{ij,l}=e_{ij,l}-e_{ij,l-1}\) is the increment of the detection error. For the \(j\)-th sample, \(\Delta e_{ij}=\big{(}\Delta e_{ij,1},\Delta e_{ij,2},\ldots,\Delta e_{ij,m_{i }}\big{)}^{T}\) is sample of random vector \(\Delta E_{ij}=\big{(}E_{ij,1}-E_{ij,0},E_{ij,2}-E_{ij,1},\ldots,\)\(E_{ij,m_{i}}-E_{ij,m_{i}-1}\big{)}^{T}\).