# Improved Bootstrap Method Based on RBF Neural Network for Reliability Assessment
## 3 Bootstrap Methodology and Its Improvement
### Assessment of Reliability Indicators  
After obtaining the expanded sample \(X^{*}(k)\) based on the maximum likelihood estimation of the two parameters of the Weibull distribution, the likelihood function \(L_{k}\) for the shape parameter \(m\) and the scale parameter \(\eta\) is calculated according to the PDF in Equation (1), as follows:  
\[L_{k}(\hat{\eta}_{k}^{*},\hat{m}_{k}^{*})=\prod_{i=1}^{n}f(x_{ki}^{*};\hat{ \eta}_{k}^{*},\hat{m}_{k}^{*}), \tag{12}\]  
where \(\hat{m}_{k}^{*}\) and \(\hat{\eta}_{k}^{*}\) are the shape parameter estimate and scale parameter estimate for the first \(k\) entries of the expanded sample, respectively. Applying the logarithmic function to Equation (12) yields the log-likelihood function. \[ \begin{array}{l}l(\hat{\eta}_{k}^{*},\hat{m}_{k}^{*})=\ln L(\hat{\eta}_{k}^ {*},\hat{m}_{k}^{*})=\sum\limits_{i=1}^{n}\ln f(x_{ki}^{*};\hat{\eta}_{k}^{*},\hat{m}_{k}^{*})\\ =\sum\limits_{i=1}^{n}\ln\hat{m}_{k}^{*}-\ln\hat{\eta}_{k}^{*}+(\hat{m}_{k}^{* }-1)\ln x_{ki}^{*}-(\hat{m}_{k}^{*}-1)\ln\hat{\eta}_{k}^{*}-\left(\frac{x_{ki} ^{*}}{\hat{\eta}_{k}^{*}}\right)^{\hat{m}_{k}^{*}}\end{array}  \tag{13}\]  
The partial derivatives of the parameters \(\hat{m}_{k}^{*}\) and \(\hat{\eta}_{k}^{*}\) in Equation (13) are equal to 0. This results in two systems of equations, as follows. \[\left\{\begin{array}{l}\frac{\partial l}{\partial\hat{\eta}_{k}^{*}}=-\frac {n}{\hat{\eta}_{k}^{*}}+\hat{m}_{k}^{*}\sum\limits_{i=1}^{n}\frac{x_{ki}^{*} \hat{\eta}_{k}^{*}}{\hat{\eta}_{k}^{*}+1}=0\\ \frac{\partial l}{\partial\hat{m}_{k}^{*}}=\frac{n}{\hat{m}_{k}^{*}}+\sum \limits_{i=1}^{n}\ln x_{ki}^{*}-n\ln\hat{\eta}  
Substituting Equation (1) into Equation (15) yields  
\[MTBF_{k}=\int_{0}^{\infty}t\cdot\frac{\hat{m}_{k}^{*}}{\hat{\eta}_{k}^{*}}\bigg{(} \frac{t}{\hat{\eta}_{k}^{*}}\bigg{)}^{m-1}e^{-(t/\hat{\eta}_{k}^{*})\hat{m}_{k}^ {*}}dt. \tag{16}\]  
Using variable substitution and the properties of the Gamma function, this integral can be simplified. The Gamma function is defined as follows:  
\[\Gamma(n)=\int_{0}^{\infty}x^{n-1}e^{-x}dx. \tag{17}\]  
Ultimately, the \(MTBF\) is  
\[MTBF_{k}=\hat{\eta}_{k}^{*}\Gamma\bigg{(}1+\frac{1}{\hat{m}_{k}^{*}}\bigg{)}. \tag{18}\]  
To ensure the accuracy of interval estimation, the method of bias correction is employed to estimate the confidence intervals. The center point of the confidence interval is modified by calculating the deviation between the original and expanded samples. The normal quantile corresponding to the position of the original sample in the cumulative distribution function of the expanded sample distribution, \(z_{0}\), is calculated as  
\[z_{0}=\Phi^{-1}\Bigg{(}\frac{1}{N}\sum_{k=1}^{N}I\big{(}\hat{\theta}_{k}^{*}< \hat{\theta}\big{)}\Bigg{)}, \tag{19}\]  
where \(\Phi^{-1}\) denotes the inverse function of the cumulative distribution function of the standard normal distribution, i.e.,  
\[\Phi^{-1}=\bigg{(}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}dt\bigg{)} ^{-1}; \tag{20}\]  
\(I\) denotes the indicator function; \(N\) denotes the number of expanded samples; \(\hat{\theta}=[\hat{\eta},\hat{m},MTBF]\) denotes the parameter estimates of the original sample; and \(\hat{\theta}_{k}^{*}=[\hat{\eta}_{k}^{*},\hat{m}_{k}^{*},MTBF_{k}]\) denotes the parameter estimates of the \(k\)th expansion sample. The values of the parameter distributions of the expanded samples may not only be biased but also asymmetric, meaning that the width of the confidence intervals may need to be skewed. The acceleration value \(a\) is used to modify the shape of the confidence interval and ensure that it adequately covers the true parameter values.