# Fiducial Lower Confidence Limit of Reliability for a Power Distribution System
## Materials and Methods
### Fiducial LCL of Reliability for a Weibull-Type Component  
In this subsection, we use the fiducial method to derive the LCL of reliability for a Weibull-type component. Let \(T_{1}\), \(T_{2}\), \(\cdots\), \(T_{n}\) be a random sample from the Weibull distribution. The cumulative distribution function of the Weibull distribution is  
\[F_{T}(t)=1-\exp\big{\{}-(t/\eta)^{m}\big{\}},\ t>0, \tag{6}\]  
where \(\eta>0\) and \(m>0\) are called the scale parameter and shape parameter, respectively. It is difficult to derive the estimators of \(\eta\) and \(m\) directly. In general, \(T_{j}\) is transformed to \(X_{j}\) by \(X_{j}=\ln T_{j}\), and then \(X_{j}\) follows an extreme value distribution with \(\sigma=1/m\) and \(\mu=\ln\eta,j=1,2,\cdots,n\). The common cumulative distribution function of \(X_{j}\), \(j=1,2,\cdots,n\) is  
\[F_{X_{j}}(x_{j})=1-\exp\bigg{\{}-e^{\frac{x_{j}-\mu}{\sigma}}\bigg{\}}. \tag{7}\]  
Let the following be true:  
\[W_{j}=\frac{X_{j}-\mu}{\sigma}=m\big{(}\ln T_{j}-\ln\eta\big{)},\ j=1,2, \cdots,n. \tag{8}\]  
Then, \(W_{j}\) follows a common standard extreme value distribution which is completely known, and the distribution of their function is also completely known. Thus, we treat the function of \(W_{j}\) as the pivotal quantity. Denote the following:  
\[\overline{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i},\ S^{2}=\frac{1}{n}\sum_{i=1}^{n} \left(X_{i}-\overline{X}\right)^{2},\overline{W}=\frac{1}{n}\sum_{i=1}^{n}W_ {i},\ V^{2}=\frac{1}{n}\sum_{i=1}^{n}\left(W_{i}-\overline{W}\right)^{2}. \tag{9}\]  
These statistics have relations as follows:  
\[\mu=\overline{X}-\frac{\overline{W}}{V}S,\sigma=\frac{S}{V},\eta=\exp\bigg{\{} \overline{X}-\frac{\overline{W}}{V}S\bigg{\}},m=\frac{V}{S}. \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Components** & **Estimators** & **Lower Confidence Limits** \\ \hline Voltage Detector & 0.9999789 & 0.9999736 \\ \hline Current Detector & 0.9999558 & 0.9999446 \\ \hline Sense Resistor & 0.9999644 & 0.9999553 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Estimators and LCLs of reliability for the exponential-type components (\(t_{0}\) = 87,600 h). Then, parameters \(\eta\) and \(m\) can be expressed as  
\[\eta=\exp\biggl{\{}\overline{X}-\frac{E_{1}}{E_{2}}S\biggr{\}},\ m=\frac{E_{2}}{S}, \tag{11}\]  
where \(E_{1}\) and \(E_{2}\) are pivotal quantities which have the same distributions as \(\overline{W}\) and \(V\), respectively. However, it is difficult to derive the explicit formula of the probability density for these two distributions. Instead, the random numbers which follow a standard extreme value distribution are easily generated by Monte Carlo simulation. Then, we treated the sample mean and the sample standard deviation as \(E_{1}\) and \(E_{2}\). The simulated values of parameters \(\eta\) and \(m\) were derived by Equation (11). By repeating this sampling method, we could get many pairs of \(\eta\) and \(m\). By averaging the sample, the estimators of the parameters were derived. The CIs were obtained by finding the sample quantiles of the simulated parameters. Here, the quantity of interest is the reliability, which can be expressed as  
\[R(t)=\exp\bigl{\{}-(t/\eta)^{m}\bigr{\}}=\exp\Biggl{\{}-\biggl{(}t\exp\biggl{\{} \frac{E_{1}}{E_{2}}S-\overline{X}\biggr{\}}\biggr{)}^{\frac{E_{2}}{S}}\Biggr{\}}. \tag{12}\]  
Then, the estimator and the LCL of reliability with a confidence level could be derived. The fiducial algorithm for the CIs of parameters \(\eta\) and \(m\) and the LCL of reliability based on a Weibull distribution is shown in Algorithm 2. ```
Input: \(t_{w}\): the lifetime of the Weibull-type component; \(1-\alpha\): the confidence level; \(t\): time;  M: the sample size of simulated pivotal quantity;  rep: the number of simulation trials. Output: The CIs of parameters \(\eta\) and \(m\) and the LCL of reliability. \(n\leftarrow\) sample size of \(t_{w}\). \(x\leftarrow\)-logarithm of \(t_{w}\). xbar \(\leftarrow\) mean of \(x\). s\(\leftarrow\) standard deviation of \(x\).