# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 1 Introduction  
Reliability refers to the ability of a product to fulfill a particular function within a predetermined period and under specified conditions. It is an important index that is used to determine the performance of a product. With the advancements in manufacturing technology, the reliability of computer numerical control (CNC) machine tools and other equipment is continuously improving, and failure data are scarce. Therefore, conventional approaches based on large sample data have limited applicability in current reliability research. The accurate estimation of the reliability index of a product using limited sample data remains a key challenge in reliability research. Currently, two main methods exist to address the problem of an insufficient sample size in reliability assessment. The first approach uses information fusion methods to fuse multiple sources of a priori information to increase the information available for assessment, thereby achieving a higher parameter accuracy. The Bayes method is an example and has been widely used in recent years for parameter estimation with the Weibull distribution [1; 2; 3]. Although it yields accurate estimation results using fewer samples, it requires a greater amount of a priori information and is affected by subjective factors. The second approach expands small data samples into large data samples. This method can also achieve accurate estimation results using fewer samples; however, the a priori information requirements are higher, and the influence of subjective factors is non-negligible. The bootstrap method is a typical example of this approach, and it is favored by scholars due to its simplicity and convenience. However, it completely relies onthe original samples, and, if these samples are not sufficiently informative, the estimation results will have a large error. Therefore, several researchers have studied and improved the bootstrap method, which was originally proposed by [PERSON] [4]. This body of work can be categorized into advancements in methodological accuracy, application to limited datasets, and refinements for enhanced sample representation. Firstly, concerning methodological accuracy, [PERSON] et al. [5] leveraged the bootstrap method for reliability estimation and analyzed the relationship between estimation accuracy and confidence level. They established that the bootstrap method attains a higher accuracy at the 95% confidence level. Building on the bootstrap's foundational use, [PERSON] et al. [6] integrated the bootstrap information criterion with bootstrap resampling to estimate reliability from limited subsample data, demonstrating the method's effectiveness even with small subsamples. Secondly, addressing the challenges of using the bootstrap method with limited data, [PERSON] et al. [7] proposed an improved bootstrap approach that ensures that expanded samples remain within the mean error of the original samples, thereby maintaining result reliability without altering the probability distribution. Similarly, [PERSON] et al. [8] developed an enhanced Bayes bootstrap method that applies an interpolation method to construct a neighborhood function, facilitating the expansion of the original sample size. Lastly, on the front of sample representation refinement, [PERSON] et al. [9] tackled the issue of large deviations between the empirical distribution function of original samples and the actual distribution. They employed a B-spline function to derive an empirical distribution more suited for sampling, which proved to meet the accuracy requirements of engineering applications. Additionally, [PERSON] et al. [10] introduced a bootstrap data expansion technique using the radial basis neural network for assessing small-sample reliability data, validating that the sample distribution characteristics closely mirror the actual distribution.