## 5 Parameter Estimation  
Supposing there are \(N\) samples, and each sample is measured \(M\) times and \(X_{k,i}(t_{j})\) is the kth measured parameter of the ith sample of the product at a time \(t_{j}\), where \(i=1,\cdots,N\),\(j=1,\cdots,M\), and \(k=1,2\). Then the measurement data of the product's degradation parameters are \(X_{k,i}(t_{j})=\left\{\left(x_{1,1}(t_{j}),x_{2,1}(t_{j})\right),\cdots,\left( x_{1,N}(t_{j}),x_{2,N}(t_{j})\right)\right\}\) and \(X_{k,i}(t_{0})=0\). If the degradation increment of the ith sample of the first performance in the time interval \(\left[t_{j-1},t_{j}\right]\) follows the Wiener process, then its likelihood function can be expressed as:  
\[L(\Delta X_{1}|\mu,\sigma_{\lambda},\sigma)=\prod_{i=1}^{N}\prod_{j=1}^{M} \frac{1}{\sqrt{2\pi\left(\sigma^{2}\Delta t_{ij}+\sigma_{\lambda}^{2}\right) \Delta t_{ij}}}exp\left\{-\frac{\left(\Delta x_{1 ij}-\mu\Delta t_{ij}\right)^ {2}}{2\left(\sigma^{2}\Delta t_{ij}+\sigma_{\lambda}^{2}\right)\Delta t_{ij} }\right\} \tag{21}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Copula Function** & \(\mathbf{C(\mu\
u\
u\theta)}\) & \(\mathbf{c(\mu\
u\
u\theta)}\) \\ \hline Frank & \(-\frac{1}{2}ln\left(1+\frac{\left(\mathrm{e}^{-\mu\theta}-1\right)\left( \mathrm{e}^{-\mu\theta}-1\right)}{\mathrm{e}^{-\mu}-1}\right)\) & \(\frac{-\theta\mathrm{e}^{-\mu\theta}(\mathrm{e}^{-\mu\theta}-1)}{\left(\left( \mathrm{e}^{-\mu}-1\right)+\left(\mathrm{e  
If the degradation increment of the ith sample of the second performance in the time interval \(\left[t_{j-1},t_{j}\right]\) obeys the Gamma process, its likelihood function is as follows:  
\[L(\Delta X_{2}|\alpha,\kappa,r)=\prod\limits_{i=1}^{N}\prod\limits_{j=1}^{M} \frac{\Gamma\big{(}\alpha\Delta t_{ij}+\kappa\big{)}}{\Gamma\big{(}\alpha\Delta t _{ij}\big{)}\Gamma(\kappa)}\cdot\frac{(r)^{\kappa}\Delta x_{2 ij}{}^{\alpha \Delta t_{ij}-1}}{\big{(}r+\Delta x_{2 ij}\big{)}^{\alpha\Delta t_{ij}+\kappa}} \tag{22}\]  
According to the feature of the Copula function, its joint probability density function \(f(\Delta X_{1},\Delta X_{2};\theta)\) can be expressed as:  
\[f(\Delta X_{1}(t),\Delta X_{2}(t);\theta)=c(F(\Delta X_{1}(t)),F(\Delta X_{2} (t));\theta)\cdot f_{1}(\Delta X_{1}(t))\cdot f_{2}(\Delta X_{2}(t)) \tag{23}\]  
Then, the log-likelihood function of the joint distribution of the product's degradation can be obtained from Equation (23) as follows:  
\[ \begin{array}{rl}lnL\big{(}F_{\Delta X_{1}},F_{\Delta X_{2}};\theta\big{)}& =\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}[lnc(F(\Delta X_{1}(\Delta t)),F( \Delta X_{2}(\Delta t));\theta)+lnf_{1}(\Delta X_{1}(\Delta t))+lnf_{2}( \Delta X_{2}(\Delta t))]\\ &=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{M}\left[\begin{array}{c}lnc\left( \Phi\left(\frac{\Delta x_{1}-\mu\Delta t_{ij}}{\sqrt{\big{(}e^{2}Al_{ij}+c_{ \Delta}^{2}\big{)}\Delta t_{ij}}}\right),F_{2 at,2x}\left(\frac{\kappa\Delta x _{2}}{\pi\Delta t_{ij}}\right);\theta\right)+\\ ln\left(\frac{1}{\sqrt{2\pi\big{(}e^{2}Al_{ij}+c_{\Delta}^{2}\big{)}\Delta t_{ ij}}}\exp\left\{-\frac{\big{(}\Delta x_{1}-\mu\Delta t_{ij}\big{)}^{2}}{2 \big{(}e^{2}Al_{ij}+c_{\Delta}^{2}\big{)}Al_{ij}}\right\}\right)\\ &\quad+ln\left(\frac{\Gamma\big{(}\alpha\Delta t_{ij}+\kappa\big{)}}{\Gamma \big{(}\alpha\Delta t_{ij}\big{)}\Gamma(\kappa)}\cdot\frac{(r)^{\kappa}\Delta x _{2 ij}{}^{\alpha\Delta t_{ij}-1}}{\big{(}r+\Delta x_{2 ij}\big{)}^{\alpha\Delta t _{ij}+\kappa}}\right)\end{array}\right]\end{array}  \tag{24}\]  
Among them, \(c(u_{1},\cdots,u_{n},\cdots,u_{N})=\frac{\partial C(u_{1},\cdots,u_{n},\cdots, u_{N})}{\partial u_{1}\cdots\partial u_{n}\cdots\partial u_{N}}\), and \(f(\cdot)\) is the PDF of the marginal distribution \(F(\cdot)\). From Equations (23)-(25), we can find that there are multiple unknown parameters in the likelihood function, and it is difficult to estimate parameters by the maximum likelihood estimation method only. Thus, the procedure of the two-step method is as shown in Figure 2.  
Figure 2: Two-stage estimation method description.