## 2 Reliability Assessment Method Based on Fusion of Simulation and Test Data
### Life Reliability Assessment Method for Zero-Failure Data
#### 2.1.2 Calculation of Life Confidence Limits  
In this study, the partition distribution curve method [25] was used to obtain the parameter estimation of the distribution function. This method is based on the estimation results for \(p_{1},\cdots,p_{m}\). The life distribution function was converted into a linear regression model, and the estimated value \(\hat{\theta}\) of the design life distribution parameter \(\theta\) was obtained using the least-squares method or the minimal \(\chi^{2}\) estimation method [26]. The calculation methods for the commonly used distributions in engineering are as follows.  
1. Normal distribution  
For a normal distribution \(F_{\theta}(t)=\Phi\big{(}\frac{t-\mu}{\eta}\big{)},t>0,\mu\in R,\sigma^{2}>0, \theta=\big{(}\mu,\sigma^{2}\big{)}\), the following linear regression model can be developed:  
\[t_{i}=\mu+\sigma\cdot\Phi^{-1}(\hat{p}_{i}),\ \ i=1,\ \...,\ \ m \tag{9}\]  
where \(\Phi^{-1}\) is the inverse function of the standard normal distribution function, taking,  
\[y=\left[\begin{array}{c}t_{1}\\ t_{2}\\ \vdots\\ t_{m}\end{array}\right],\ \ A=\left[\begin{array}{cc}1&\Phi^{-1}(\hat{p}_{1})\\ 1&\Phi^{-1}(\hat{p}_{2})\\ \vdots&\vdots\\ 1&\Phi^{-1}(\hat{p}_{m})\end{array}\right],\beta=\left[\begin{array}{c}\mu \\ \sigma\end{array}\right] \tag{10}\]  
The least-squares estimate of the parameter \(\beta\) is,  
\[\hat{\beta}=\left[\begin{array}{c}\hat{\mu}\\ \hat{\sigma}\end{array}\right]=\left(A^{T}A\right)^{-1}A^{T}y \tag{11}\]  
According to the given reliability \(R\), the one-sided confidence lower bound formula [27] of the Q-percentile life \(t_{R}\) with a confidence coefficient of \(\gamma\) is,  
\[t_{RL}=\overline{t}-(t_{R}-t_{R_{0}})\sigma \tag{12}\]  
where \(\overline{t}=\frac{1}{n}\sum\limits_{i=1}^{n}t_{i},\ t_{R}=\Phi^{-1}(R)\), \(t_{R0}=\Phi^{-1}(R_{0})\) and \(R_{0}=\left(1-\gamma\right)^{1/n}\), and the standard deviation \(\sigma\) is typically determined by engineering experience, but in this case it can be replaced with the fitted result \(\hat{\sigma}\) instead.  
2. Weibull distribution For the Weibull distribution \(F_{\theta}(t)=1-\exp\left\{-\left(\frac{t}{\eta}\right)^{k}\right\}\), where \(\theta=(\mu,\sigma)\), it is known that \(\ln\frac{1}{1-F_{\theta}(t)}=\left(\frac{t}{\eta}\right)^{k}\); thus, the following linear regression equation is obtained:  
\[\ln t_{i}=a+b\cdot\ln\ln\frac{1}{1-\hat{p}_{i}}+\epsilon_{i},i=1,...,m \tag{13}\]  
where \(a=\ln\eta\) and \(b=\frac{1}{k}\). Note that,  
\[y=\left[\begin{array}{c}\ln t_{1}\\ \ln t_{2}\\ \vdots\\ \ln t_{m}\end{array}\right],\;\;A=\left[\begin{array}{cc}1&\ln\ln\frac{1}{ 1-\hat{p}_{1}}\\ 1&\ln\ln\frac{1}{1-\hat{p}_{2}}\\ \vdots&\vdots\\ 1&\ln\ln\frac{1}{1-\hat{p}_{m}}\end{array}\right],\beta=\left[\begin{array}[] {c}a\\ b\end{array}\right] \tag{14}\]  
Then, the least-squares estimate of the parameter \(\beta\) is,  
\[\hat{\beta}=\left[\begin{array}{c}\hat{a}\\ \hat{b}\end{array}\right]=\left(A^{T}A\right)^{-1}A^{T}y \tag{15}\]  
When,  
\[R\geq\exp\left\{\ln(1-\gamma)\exp\left\{\frac{\sum_{i=1}^{n}t_{i}^{A_{0}}\ln t _{i}^{A_{0}}}{\sum_{i=1}^{n}t_{i}^{A_{0}}}-\ln\sum_{i=1}^{n}t_{i}^{A_{0}} \right\}\right\} \tag{16}\]  
In which the reliability \(R\) being given, the one-sided confidence lower bound formula [27] of the Q-percentile life \(t_{R}\) with a confidence coefficient of \(\gamma\) is,  
\[t_{RL}=\left[\frac{\ln R}{\ln(1-\gamma)}{\sum_{i=1}^{n}t_{i}^{k}}\right]^{1/k} \tag{17}\]  
The shape parameter \(k\), which is usually determined by engineering experience, can be replaced with the fitting results \(\hat{k}\).