## 4 Numerical Examples  
### Model Verification  
To demonstrate the effectiveness and applicability of the above model, this section compares some degradation methods with an illustrative example. For simplicity, the proposed model in this paper in Equation (17) is referred to as \(M_{0}\).  
\[M_{0}:Y(t|S_{k})=\lambda_{k}\Lambda(t)+(\sigma_{B})_{k}B(\Lambda(t))+\sigma_{ \varepsilon}\varepsilon \tag{29}\]  
where \(\lambda_{k}=\eta\exp(-b/S_{k})\), \(\eta\sim N\Big{(}\mu_{\eta},\sigma_{\eta}^{2}\Big{)}\), \(\big{(}\sigma_{B}^{2}\big{)}_{k}=\kappa\varsigma_{k}\), \(\sigma_{\varepsilon}\varepsilon\sim N\big{(}0,\sigma_{\varepsilon}^{2}\big{)}\).  
If the acceleration factor constant principle is not considered, then \(M_{0}\) becomes a nonlinear Wiener process model whose diffusion parameter \(\sigma_{B}\) is fixed that has no relationship with the applied stress, which is referred to \(M_{1}\).  
\[M_{1}:Y(t|S_{k})=\lambda_{k}\Lambda(t)+(\sigma_{B})_{k}B(\Lambda(t))+\sigma_{ \varepsilon}\varepsilon \tag{30}\]  
where \(\lambda_{k}=\eta\exp(-b/S_{k})\), \(\eta\sim N\Big{(}\mu_{\eta},\sigma_{\eta}^{2}\Big{)}\), \(\sigma_{\varepsilon}\varepsilon\sim N\big{(}0,\sigma_{\varepsilon}^{2}\big{)}\).  
If the measurement errors are not considered, then \(M_{0}\) becomes a nonlinear Wiener process model without measurement errors which is referred to \(M_{2}\).  
\[M_{2}:Y(t|S_{k})=\lambda_{k}\Lambda(t)+(\sigma_{B})_{k}B(\Lambda(t)) \tag{31}\]  
where \(\lambda_{k}=\eta\exp(-b/S_{k})\), \(\eta\sim N\Big{(}\mu_{\eta},\sigma_{\eta}^{2}\Big{)}\).  
The fits of these different Wiener process models were compared by the Akaike Information Criterion (AIC). AIC handles the trade-off between the goodness of model fit and the complexity of the model, which is defined as  
\[AIC=2m-2\ell \tag{32}\]  
In Equation (32), the number of unknown parameters is \(m\) and the maximized value of the log-likelihood function of the estimated model is \(\ell\). The model with the lowest AIC values in several potential models would be selected as the best-fit one.  
Taking the example of the constant accelerated degradation data of light-emitting diodes (LED) in [26], the data contain three accelerated levels of thermal stress, \(25\,^{\circ}\)C, \(65\,^{\circ}\)C, and \(105\,^{\circ}\)C. At each accelerated thermal stress, there were 25 LEDs whose light intensity was recorded at 29 inspection times. Supposing the initial value is 1, LED fails when the relative luminosity drops to 0.5. Thus, 50% of the initial luminosity is the failure threshold. The degradation paths of light intensity of LEDs are shown in Figure 1.  
According to the degradation trajectory of the 75 LEDs, let \(\Lambda(t)=t^{0.5}\) for simplicity. The degradation results of LEDs mentioned above are listed in Table 2 after comparing the applicability of different models mentioned above, and we can find that the model \(M_{0}\) has the lowest \(AIC\) and the highest \(\ell\) compared with other models. It shows that the raised model in this paper has a better model goodness-of-fit than the others.  
Thus, it is necessary to consider the nonlinearity, acceleration factor constant principle, unit-to-unit variability, and measurement errors when modeling the degradation paths.