## 2 Theoretical Background
### Life Assessment Method Based on IG Process for Accelerated Degradation Test  
Without considering the random factors of the product degradation path, assuming that \(\{Y(t),t\geq 0\}\) represents the product degradation and that \(Y(t)\) conforms to the IG process, then \(Y(t)\) has the following three characteristics:  
1. The value of \(Y(0)\) is always 0. 2. For any \(0<t_{1}<t_{2}<t_{3}<t_{4}\), \(Y(t_{4})-Y(t_{3})\) and \(Y(t_{2})-Y(t_{1})\) are independent degradation increments. 3. For any \(0\leq s<t\), \(Y(t)-Y(s)\) follows the IG process with mean \(\mu\Delta\Lambda\) and variance \(\mu^{3}\Delta\Lambda/\lambda\), which is denoted as \(Y(t)-Y(s)\sim IG\left(\mu\Delta\Lambda,\lambda(\Delta\Lambda)^{2}\right)\). Here, \(\mu\) and \(\lambda\) are constants. \(\Delta\Lambda=\Lambda(t)-\Lambda(s)\), \(\Lambda(0)=0\), and \(\Lambda(\cdot)\) is the monotone increasing function of time \(t\). Based on the above three characteristics, the IG process model for \(Y(t)\) can be established. Then, the MLE method is used to calculate the model parameters \(\mathbf{\theta}=(\mu,\lambda)^{\prime}\), thereby obtaining its reliability function. The specific details are as follows: It is assumed that the degradation amount \(Y(t)\) follows the IG process with mean \(\mu\Lambda(t)\) and variance \(\mu^{3}\Lambda(t)/\lambda\), which is denoted as \(Y(t)\sim IG\left(\mu\Lambda(t),\lambda\Lambda^{2}(t)\right)\). Here, \(\mu\) is the degradation rate. \(\lambda\) has no direct physical significance. Under the stress level \(S_{i}\), \(\lambda\) is constant, and the mean \(\mu\Lambda(t)\) and variance \(\mu^{3}\Lambda(t)/\lambda\) of \(Y(t)\) are increasing. Therefore, let \(\mu=h(S)\), where  
Figure 1: The SDSADT stress profile diagram. \(h(S)\) is a function of the stress level and the product performance degradation. Then, the probability density function (PDF) of \(Y(t)\) is as follows:  
\[f(y(t)|h(x),\lambda)=\sqrt{\frac{\lambda\Lambda^{2}(t)}{2\pi y(t)^{3}}}\exp\left[ -\frac{\lambda(y(t)-h(x)\Lambda(t))^{2}}{2h(x)^{2}y(t)}\right] \tag{4}\]  
It is assumed that under stress level \(S_{i}\), the degradation amount of product \(j\) at the time \(t_{l}\) is \(\Upsilon_{ij}(t_{l}),(i=1,2,\ldots,k;j=1,2,\ldots,n;l=1,2,\ldots,m_{i})\). Here, \(k\) represents the number of stress levels, \(n\) represents the number of test products, and \(m_{i}\) represents the detection number under stress level \(S_{i}\). Then, the degradation increment of products is \(\Delta y_{ij,l}=Y_{ij,l}-Y_{ij,l-1}\), and the increment of the monotone increasing function is \(\Lambda_{ij,l}(t)=\Lambda\Big{(}t_{ij,l}\Big{)}-\Lambda\Big{(}t_{ij,l-1}\Big{)}\). Therefore, the likelihood function of the accelerated degradation model based on the IG process is as follows:  
\[L(\mathbf{\theta})=\prod_{j=1}^{n}\prod_{i=1}^{k}\prod_{l=1}^{m_{i}}f\Big{(}\Delta y _{ij,l};\mathbf{\theta}\Big{)}=\prod_{j=1}^{n}\prod_{i=1}^{k}\prod_{l=1}^{m_{i}} \left(\sqrt{\frac{\lambda\Big{(}\Lambda_{ij,l}\Big{)}^{2}}{2\pi\Delta y_{ij,l} ^{2}}}\times\exp\left[-\frac{\lambda\Big{(}\Delta y_{ij,l}-e^{\alpha_{0}+ \alpha_{1}x_{i}}\Lambda_{ij,l}\Big{)}^{2}}{2e^{2(\alpha_{0}+\alpha_{1}x_{i})} \Delta y_{ij,l}}\right]\right) \tag{5}\]  
where \(\Lambda(\cdot)\) is a monotone increasing function of time \(t\), indicating that the cumulative degradation effect of the product is proportional to time. The value of \(\Lambda(\cdot)\) can be determined based on the personal experience of reliability engineers, knowledge of failure physics, handbooks, etc. [24]. In this article, \(\Lambda(t)=t\). This provides a simplified linear accumulation assumption for the model, where the passage of time directly reflects the increase in the cumulative degradation effect. \(\mathbf{\theta}\) is the unknown parameter.