## 3 Proposed Method
### Parameter Estimation and Storage Life Assessment  
Considering the complexity of the SDRD model and the absence of an explicit expression for the likelihood function, traditional parameter estimation methods are challenging to meet the requirements. MC is a feasible method for approximating the solution of a problem through random sampling and extensive simulations. The core idea of this method is to generate a large number of random samples and perform statistical analysis based on these samples to obtain estimated values for the target parameters. Therefore, this article uses the MC method to estimate the unknown parameters in the IG process model, which can effectively enhance calculation accuracy and efficiency. The algorithm flow is shown in Figure 2, and the specific details are as follows:  
First, let the function of \(\Delta y_{ij}\) and \(\Delta e_{ij}\) be denoted as \(g\left(\Delta y_{ij},\Delta e_{ij}\right)\), and its formula as follows:  
\[g\left(\Delta y_{ij},\Delta e_{ij}\right)=\left\{ \begin{array}{cl}f_{\Delta X _{ij}}\left(\Delta y_{ij}-\Delta e_{ij};\alpha_{0},\alpha_{1},\delta,\gamma \right),&\Delta e_{ij,l}<\Delta y_{ij,l},l=1,2,\ldots m_{i}\\ 0,&otherwise\end{array} \right. \tag{21}\]Next, the likelihood function (19) can be understood as the expectation of the function \(g\Big{(}\Delta\mathbf{y}_{ij;}\,\Delta\mathbf{e}_{ij}\Big{)}\), that is  
\[ \begin{array}{ll}L_{ij}\Big{(}\Delta\mathbf{y}_{ij;}\,\mathbf{\theta}\Big{)}&=\int \limits_{\Theta}f_{\Delta\mathbf{x}_{ij}}\Big{(}\Delta\mathbf{y}_{ij}-\Delta\mathbf{e}_{ij; }\,\alpha_{0},\alpha_{1},\delta,\gamma\Big{)}f_{\Delta\mathbf{e}_{ij}}\big{(} \Delta\mathbf{e}_{ij;}\,\sigma_{x}^{2}\big{)}\,d\Delta\mathbf{e}_{ij}\\ &=\int\limits_{\Theta}g\Big{(}\Delta\mathbf{y}_{ij;}\,\Delta\mathbf{e}_{ij}\Big{)}f_{ \Delta\mathbf{e}_{ij}}\big{(}\Delta\mathbf{e}_{ij}\big{)}\,d\Delta\mathbf{e}_{ij}\\ &=\int\limits_{\kappa^{m}_{ij}}g\Big{(}\Delta\mathbf{y}_{ij;}\,\Delta\mathbf{e}_{ij} \Big{)}f_{\Delta\mathbf{e}_{ij}}\big{(}\Delta\mathbf{e}_{ij}\big{)}\,d\Delta\mathbf{e}_{ ij}\\ &=E\Big{[}g\Big{(}\Delta\mathbf{y}_{ij;}\,\Delta\mathbf{e}_{ij}\Big{)}\Big{]}\end{array}  \tag{22}\]  
where \(E(\cdot)\) represents the expectation operation. Finally, generate a group of multivariate random samples based on \(\Delta\mathbf{E}_{ij}\), then calculate the average, and subsequently use the MC method to calculate the MLE of the parameters. The specific process is as follows:  
Step 1: Set the value of the parameter \(\alpha_{0},\alpha_{1},\delta,\gamma,\sigma_{x}^{2}\). Step 2: Generate \(N\) sets of random samples \(\mathbf{e}_{ij}^{(k)}=\Big{(}e_{ij;0}^{(k)},e_{ij;1}^{(k)},e_{ij;2}^{(k)},\ldots,e_ {ij;m_{i}}^{(k)}\Big{)},k=1,2,\ldots,N\) from the normal distribution with mean 0 and variance \(\sigma_{x}^{2}\). Step 3: Calculate the error increment \(\Delta e_{ij}^{(k)}=e_{ij;l}^{(k)}-e_{ij;l-1}^{(k)}\). Step 4: \(\Delta e_{ij}^{(k)}=e_{ij;l}^{(k)}-e_{ij;l-1}^{(k)}\) is the sample of the multivariate normal random variable \(\Delta\mathbf{E}_{ij}\); obtain the likelihood function based on the geometric mean of the following formula. \[L_{ij}\Big{(}\Delta\mathbf{y}_{ij;}\,\mathbf{\theta}\Big{)}\approx\frac{1}{N}\sum \limits_{k=1}^{N}g\Big{(}\Delta\mathbf{y}_{ij;}\,\Delta\mathbf{e}_{ij}^{k}\Big{)} \tag{23}\]  
Step 5: Assuming \(N\) is the specified number of simulation iterations, repeat Step 2 to Step 4 \(N\) times to obtain \(N\) likelihood function values \(L_{ij}^{k}(k=1,2,\ldots,N)\). The maximum value of the likelihood function is the MLE of the parameter, that is  
\[\Big{(}\hat{\alpha}_{0},\hat{\alpha}_{1},\hat{\delta},\hat{\gamma},\hat{\sigma }_{x}^{2}\Big{)}=\max\Big{\{}L_{ij}^{k}\Big{(}\Delta\mathbf{y}_{ij;}\,\mathbf{\theta} \Big{)},k=1,2,\ldots,N\Big{\}} \tag{24}\]  
Step 6: Approximate the conditional distribution \([Y(t)|\lambda]\) using the normal distribution and marginalize \(\lambda\) to estimate the product life \(T_{D_{f}}\)[25].