# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 3 Bootstrap Methodology and Its Improvement  
### Bootstrap Approach  
Let \(X=[x_{1},x_{2},\ldots,x_{n}]\) be a set of random variables with a joint distribution \(F_{n}\). To estimate the overall parameter \(\theta\), it is generally possible to obtain a sample-based estimate \(\theta_{n}\). The basic concept underlying the bootstrap method is that, given \(X\), one can construct an estimate \(\hat{F}_{n}\) of \(F_{n}\) and then regenerate a set of random variables \(X^{*}=\left[x_{1}^{*},x_{2}^{*},\ldots,x_{n}^{*}\right]\) from the distribution \(\hat{F}_{n}\). If \(\hat{F}_{n}\) is the best estimate of \(F_{n}\), then the relationship between \(X\) and \(F_{n}\) is adequately represented by the relationship between \(X^{*}\) and \(\hat{F}_{n}\), where \(\hat{F}_{n}\) is called the empirical distribution function of the bootstrap method. This step can be repeated  
Figure 1: Weibull distribution probability density function (PDF). several times to obtain multiple estimates from the reconstructed data according to an estimation equation, as that for \(\theta_{n}\). The metrics for measuring the accuracy of the estimator can then be obtained (e.g., using the Bayes method). The principle of the bootstrap method is illustrated in Figure 2. and the procedure is as follows. The order statistics of the samples can be obtained by arranging the original samples \(X=[x_{1},x_{2},...,x_{n}]\) in descending order, i.e., \(x_{(1)},x_{(2)},\ldots,x_{(n)}\), where \(x_{(1)}=\min\limits_{1\leq i\leq n}x_{i}\cdots\) and \(x_{(n)}=\max\limits_{1\leq i\leq n}x_{i}\). When the parameters of the estimated distribution \(\hat{F}_{n}\) are unknown and if the value of the cumulative distribution function at \(x_{i}\) is \(F_{i}=\frac{i}{n}\), then the empirical distribution function of the original sample assuming equal probability sampling is  
\[F_{n}(x)= \begin{cases}0&x<x_{(1)}\\ k/n&x_{(k)}\leq x\leq x_{(k+1)},k=1,2,\ldots,n-1\\ 1&x>x_{(n)}\end{cases} . \tag{5}\]  
The simulation-based method for generating random samples that obey the empirical distribution function \(F_{n}(x)\) is as follows:  
1. Uniformly distributed pseudo-random numbers \(\eta\) in the interval [0, 1] are generated;
2. Let \(\beta=(n-1)\eta,i=[\beta]+1\), where \([\beta]\) is rounded down;
3. Let \(x_{F}=x_{(i)}+(\beta-i+1)\Big{(}x_{(i+1)}-x_{(i)}\Big{)}\), where \(x_{F}\) is the desired random sample. A review of existing studies and experimental simulations revealed that the resampling of the data in the bootstrap method relies on the original samples. Therefore, the random samples generated are typically not representative of the whole population, and the estimates obtained may be biased. Moreover, the bootstrap method may not be robust enough in terms of the margins (i.e., extremes) of the data distribution, because the extremes may be over-represented or under-represented in the generated random samples generated. The bootstrap method is, therefore, not reasonable when processing small subsamples of data. There are two main reasons for this. First, when \(F_{n}(x)\) is used to generate random samples, the sample values are extracted from the original sample with a medium  
Figure 2: Schematic of the bootstrap method. probability to form an expanded sample. The resulting sample's empirical function \(F_{n}(x)\), which is used to fit the head and the tail samples, is inadequate. Consequently, the samples generated according to \(F_{n}(x)\) are not satisfactorily random. Second, because the generated random samples are limited by the minimum and maximum values \([x_{(1)},x_{(n)}]\)[16] and the values of random samples can only be extracted from the head and tail samples using limited subsample data, the bootstrap method is ineffective. Moreover, as the values of the random samples can only be obtained from the range of the original samples, the samples are not adequately random [17; 18]. Therefore, this paper aims to address these two problems.