## 3 Numerical Example  
In this section, the simulation and test data fusion method is validated through a numerical example. Assume that we have the following function: \(T(x_{1},x_{2})=e^{x_{1}}+\frac{x_{2}}{e^{x_{1}}}\), where \(x_{1}\) and \(x_{2}\) are mutually independent random variables satisfying the lognormal and normal distributions, respectively, \(\ln x_{1}\sim\ N(5,\,0.2),\ x_{2}\sim\ N(34500,\,100)\). Let \(G(x_{1},x_{2})=e^{x_{1}}+\frac{x_{2}}{e^{x_{1}}}-6050\); when \(G>0\), the data are considered as failure data, and when \(G<0\), the data are considered as zero failure data.  
First, \(300\) individual points were sampled, and the sample set D was obtained using the function T. The criterion function \(G(x_{1},x_{2})\) was used to determine whether the data were  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Conditions** & **Conclusions** \\ \hline \(t_{S,R}>t_{N,RU}\) & The simulation was found to be incorrect at confidence level \(\gamma\) \\  & The simulation is more adventurous than the real situation \\  & and is dangerous for engineering applications. \\ \hline \(t_{S,R}<t_{N,RL}\) & The simulation was found to be incorrect at confidence level \(\gamma\) \\  & However, because the simulation is more conservative and \\  & safer than the real situation, it can be applied in engineering. \\ \hline \(t_{N,RL}<t_{S,R}<t_{N,RU}\) & When \(t_{N,RU}-t_{N,RL}\leq\delta\), the confidence level \(2\gamma-1\) \\  & determines that the simulation is correct, i.e., \(t_{S,R}=t_{R}\), at the \\  & given accuracy \(\delta\). \\ \hline \(t_{N,RL}<t_{S,R}<t_{N,RU}\) & When \(t_{N,RU}-t_{N,RL}>\delta\), additional testing with additional specimens is needed to determine the reliability. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Life reliability simulation test criteria.  
zero-failure data. The sample set D was used to construct the proxy model \(\widetilde{T}(x_{1},x_{2})\), which described the simulation computation process. A radial basis function neural network (RBFNN) was used to construct the proxy model. The \(\widetilde{T}\) and \(T\) are different here, indicating differences in the distributions of the simulation data and the test data.  
Second, sampling was performed \(10^{6}\) times based on Monte Carlo simulation (MCS), and with the help of the function \(T(x_{1},x_{2})\) and the proxy model \(\widetilde{T}(x_{1},x_{2})\), stable distributions of the random variables \(T\) and \(T\) were obtained, as shown Figure 1.  
Last, we determined that the reliability was \(R\) = 0.95, the true value was 6385, and the simulation prediction was 5994.  
The zero-failure data in sample set D were used as the truncated test data. The distribution of the random variable \(\widetilde{T}\) approximately followed the normal distribution, and the probability of failure at 6050 did not exceed 0.1; thus, \(q\) was 0.1. The truncated test data and the probability of failure estimated using the Bayes method are shown in Table 2.  
According to the reliability parameter estimation method presented in Section 2.1, the point \(\hat{k}\) was estimated as 281.6698 using Equation (15). Additionally, using Equation (12), we obtained the confidence level \(\gamma\) = 0.95, and the life one-sided lower confidence limit \(t_{N,RL}\) with reliability \(R\) = 0.95 was obtained as 6192.  
Since \(t_{S,R}<t_{N,RL}\), it was concluded that the predicted value obtained from the simulation at the confidence level \(\gamma\) = 0.95 was smaller than the true value; i.e., the simulation results were conservative, and the simulation data could be used for fusion. The fusion of \(t_{N,RL}\) and \(t_{S,RL}\) was performed in accordance with Equation (19). The confidence level was \(\gamma\) = 0.95, and the life one-sided lower confidence limit \(t_{T,RL}\) with reliability \(R\) = 0.98 was 6042 after the fusion.