# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 6 Conclusions  
The estimation of equipment _MTBF_ is crucial for reliability assessment and analysis. However, when the number of samples is limited, relying on traditional parameter estimation methods simulations is inadequate. Moreover, conventional parameter estimation methods such as maximum likelihood estimation typically fail to estimate the confidence intervals of the parameters. This paper proposes the use of the bootstrap method for data expansion and reliability assessment. An exponential distribution is utilized to fit right-tailed data and modify the empirical distribution function. The simulation results indicate that the range of the expanded samples generated via the modified bootstrap method increases. The randomness of the expanded samples also increases, and the accuracy of interval estimation improves. In addition, a novel data expansion method is proposed by combining the modified bootstrap method with the RBF neural network. The bias correction method is then used to estimate confidence intervals for the expanded data and improve the estimation accuracy. Through our analysis of the results, this paper proposes that the method of tail data correction using an exponential function effectively enhances the original failure data by moderately incorporating additional information on the product's reliability characteristics, based on its failure properties. This approach optimizes the raw data. Furthermore, employing the radial basis function (RBF) neural network essentially achieves a better fit of the failure data, thereby improving the accuracy of parameter estimation. Finally, the proposed method is employed for the reliability assessment of a CNC machine tool. The shape and scale parameters of the corresponding Weibull distribution are estimated to determine the _MTBF_ of the equipment. Simulation experiments show that the proposed method offers a greater improvement in the accuracy of point estimation and interval estimation than the original bootstrap and conventional parameter estimation methods. Therefore, this method has excellent applicability in engineering practice. Despite \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Point** & **Rated** & **Absolute** & **Relative** & **Confidence** & **Interval** \\  & **Estimates (h)** & **Value (h)** & **Error (h)** & **Error (\%)** & **Interval** & **Length** \\ \hline Maximum likelihood method & 1118.20 & & 118.20 & 11.82 & \(\backslash\) & \(\backslash\) \\ Bootstrap & 1114.97 & & 114.97 & 11.50 & (1099.51, 1130.47) & 30.96 \\ RBF + conventional bootstrap & 1118.32 & & 118.32 & 11.83 & (1114.37, 1121.86) & 7.49 \\ RBF +improved bootstrap & 1083.41 & & 83.41 & 8.34 & (1080.13, 1089.15) & 9.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of reliability assessment results based on _MTBF_ obtained using the different methods. our research's contributions, our work is not without limitations. The collection of CNC failure data presents significant challenges, notably due to the scarcity of available data. We compiled data from seven machines operating under ostensibly similar conditions, operating on the assumption that these conditions were identical. However, in reality, variances in operating conditions do exist. Addressing how to effectively integrate data across varying conditions represents a key area for our future investigations. Conceptualization, H.L. and H.W. ; methodology, S.S.; software, H.W. ; validation, H.L. and H.W. ; formal analysis, H.W. ; investigation, S.S.; resources, S.S.; data curation, H.W. ; writing--original draft preparation, H.W. ; writing--review and editing, H.L. and S.S. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study are available upon request from the corresponding author. The data are not publicly available due to privacy. We extend our deepest appreciation to [PERSON] for his invaluable guidance, unwavering support, and mentorship throughout this research.