# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 5 Example Analysis
The overall distribution of \(MTBF_{TR}\) is shown in Figure 9a, and the parameter distribution of \(X_{TR}*\) is shown in Figure 10a, and the parameter distribution of \(MTBF_{BR}\) is shown in Figure 10b. Figure 8: Results of the conventional bootstrap approach. (**a**) Overall distribution; (**b**) parametric distribution. The cumulative distribution function (CDF) and probability density function (PDF) are obtained using Equations (1) and (2), as shown in Figure 11.  
Figure 10: Results of the RBF + improved bootstrap method. (**a**) Overall distribution; (**b**) parametric distribution. Figure 9: Results of the RBF + conventional bootstrap method. (**a**) Overall distribution; (**b**) parametric distribution. As illustrated in Figure 11, the probability density function (PDF) of the Weibull life distribution, obtained using the RBF plus the enhanced Bootstrap method, exhibits a peak value that is comparatively higher than those obtained through other methods. This observation can be interpreted as follows:  
1. A higher peak value indicates that the life data are more concentrated around a specific time period. This suggests that the majority of components or systems are likely to fail around this point in time, demonstrating a lower variability in life spans. In other words, the lifespans of most components are expected to be relatively similar, leading to reduced uncertainty in life expectancy predictions. 2. Additionally, a higher peak value implies more accurate reliability predictions at this specific time point. Since failure events are more likely to occur near the peak, this facilitates more precise planning for maintenance, replacement cycles, and inventory management. A comparison of the _MTBF_ estimates obtained from the aforementioned methods with the manufacturer-rated _MTBF_ = 1000 h and the corresponding errors are presented in Table 4.  
Figure 11: Plots of the cumulative distribution function (CDF) and probability density function (PDF). As shown in Table 4, the estimated value of the _MTBF_ obtained using the maximum likelihood estimation method is 1118.20 h, compared with the nominal value of 1000 h, resulting in a relative error of 11.82%. The relative error of the RBF + conventional bootstrap method is 11.83%, which is almost equal to that of the maximum likelihood estimation method, indicating that the expanded samples obtained using the RBF + conventional bootstrap method are overfitted and not sufficiently random for the bootstrap method. The analysis results presented in Figures 9a and 10a reveal that correcting the tail of the empirical distribution function using an exponential distribution attenuates the proportion of large values and makes the distribution more dispersed, which is consistent with the actual life distribution of the equipment. In addition, the relative error for _MTBF_ is reduced to 8.34% from 11.50%, indicating that the proposed data expansion method improves the conventional bootstrap method. In this study, the confidence interval estimates for the different methods are obtained by combining bias correction methods. As shown in Table 4, the bootstrap method combined with the RBF neural network significantly reduces the length of the confidence intervals and improves the accuracy of the estimates. This demonstrates the effectiveness of combining RBF neural networks with the bootstrap method.