## 3 ADDT Analysis
### Failure-Time Analysis  
To derive the failure-time distribution in normal use conditions, we used two methods; closed-form expression and the Monte Carlo simulation, based on the estimated parameters for the power degradation model selected as the best for the ADDT experiment data. Using the variance-covariance matrix of estimated parameters, 100,000 degradation paths were simulated from the Monte Carlo simulation. The pseudo-failure is determined at the time that the degradation path reaches the given failure threshold (86.89%). Suppose that a specified model for degradation path \(\mathcal{D}(t)\) and the failure-time distribution can be derived by the function of the degradation model parameters. Assume that a unit fails at given time \(t\) for the degradation path to reach a given criterion \(\mathcal{D}_{f}\)  
\[\Pr(T\leq t)=F_{y}(t;x,\mathbf{\theta})=\Pr\Bigl{[}\mathcal{D}(t;x,\mathbf{\theta}) \leq\mathcal{D}_{f}\Bigr{]}. \tag{10}\]  
For a given \(\mathcal{D}_{f}\), the distribution of \(T\) depends on the distribution of the degradation model parameters \(\mathbf{\theta}\). In general, a closed-form expression exists and the life estimators can be calculated based on the nonlinear \(\mathcal{D}(t)\) with the MLE \(\mathbf{\theta}\). The \(p\)th quantile of the lifetime distribution is derived using the relationship  
\[\mathcal{D}\bigl{(}t_{p};x,\mathbf{\theta}\bigr{)}=\ \mathcal{D}_{f}-\vartheta \cdot\Phi^{-1}(p), \tag{11}\]  
and by solving the power model in terms of \(t_{p}\). In addition to the lifetime estimation using the closed-form equation, a numerical method was applied to compare the precision of the estimated results. A Monte Carlo \begin{table}
\begin{tabular}{c c c c} \hline \hline
**ADDT Model** & **Log-Likelihood** & **AIC** & **BIC** \\ \hline Exponential & 222.54 & \(-439.08\) & \(-430.25\) \\ Logarithmic & 373.49 & \(-740.99\) & \(-732.16\) \\
**Power** & **374.53** & \(\mathbf{-743.07}\) & \(\mathbf{-734.24}\) \\ Logistic & 230.38 & \(-454.75\) & \(-445.93\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stress condition for the failure mechanism. Figure 10: Fitting results of the ADDT data from the four candidate models. simulation can be an effective approach for the evaluation of \(F_{y}(t;x,\theta)\) by the large number of random sample paths generated with the assumed model. The procedure of the lifetime estimation based on the simulation algorithm is given as follows. * Step I: Generate \(N\) (large number: e.g., \(N=100,000\)) simulated degradation paths using the estimators \(\left(\hat{\beta}_{(1)},\hat{E}_{a(1)}\right),\ldots,\left(\hat{\beta}_{(N)}, \ \hat{E}_{a(N)}\right)\) of \(\left(\hat{\beta},\ \hat{E}_{a}\right)\) from a multivariate normal distribution with mean \(\hat{\theta}=\left(\hat{\beta},\ \hat{E}_{a}\right)=\left(-0.0114,\ 0.2571\right)\) and variance-covariance matrix \(\hat{\Sigma}_{\hat{\theta}}\), where we used the variance-covariance matrix of \(\hat{\theta}\equiv\left(\hat{\beta},\ \hat{E}_{a}\right)\) to simulate the random sample path. * Step II: Compute \(N\) pseudo-failure times corresponding to the \(N\) realizations of \(\left(\hat{\beta}_{(1)},\ \hat{E}_{a(1)}\right),\ldots,\left(\hat{\beta}_{(N)},\ \hat{E}_{a(N)}\right)\), where the pseudo-failure time is defined as the point where the degradation path intersects with the failure threshold. * Step III: For any desired values of \(t\), use the following function for evaluation of \(F(t)\) as \[F(t)\approx\frac{\#\ of\ simulated\ intersecting\ time\leq t}{N}.\] As the value of \(N\) increases, the precision of the estimation will improve. Using the Monte Carlo simulation, the simulated degradation paths with the given failure criterion and resulting failure-time distribution is given in Figure 11.  
The results of the lifetime estimation using both the closed-form equation and Monte Carlo simulation are compared in Table 3, where the years of use were calculated according to the average usage rate of the vehicle at 4 h per day.