## 4 Numerical Example  
A total of 58 failure data points are collected from five machining centers. Symbols of the subsystem are shown in Table 1. The failure interval time is sorted from smallest to largest, and the results are shown in Table 2.  
The improved average rank method is used to obtain the model parameters. Taking the hydraulic subsystem as an example, the calculation process is as follows. Firstly, the average rank method is used to calculate the empirical distribution function, and the results are shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Subsystem** & **Symbol** \\ \hline Main transmission subsystem & T \\ Feed subsystem & F \\ Hydraulic subsystem & H \\ Lubricating subsystem & L \\ Numerical control subsystem & NC \\ Electrical subsystem & E \\ Cooling subsystem & C \\ Auxiliary subsystem & A \\ Automatic tool changer & ATC \\ Servo subsystem & S \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols for the subsystem. Figure 3: Solution process of the maintenance model. The model parameters are calculated using the least square equations in Section 2.2. The results are \(\hat{B}=1.839\), \(\hat{A}=-14.058\), \(\hat{\beta}=1.839\), and \(\hat{\eta}=2092.333\). The results are shown in Figure 4. The rank increment is updated using the improved average rank method; afterward, the updated empirical distribution function and the updated parameters are obtained. The calculation results of the rank increment are shown in Table 4. The updated parameters are \(\hat{\beta}=1.771\), \(\hat{A}=-13.602\), \(\hat{B}=1.771\), and \(\hat{\eta}=2161.748\). The results are shown in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(j\)} & \multirow{2}{*}{**TBF/h**} & \multirow{2}{*}{**Sub  
+++ ==WARNING: Truncated because of repetitions==
-System**} \\ \cline{5-6}  & & & & & & & & \\ \hline
1 & 1.53 & E & 21 & 730.3 & NC & 41 & 1103.12 & NC \\
2 & 19.18 & S & 22 & 734.14 & H & 42 & 1129.21 & T \\
3 & 56.77 & C & 23 & 735.67 & F & 43 & 1139.95 & NC \\  
++  
Figure 4: Average rank method results. Figure 5: Improved average rank method results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{j}\) & \(\mathbf{i}\) & \(\mathbf{t}_{i}\) & \(\mathbf{\Delta}\mathbf{A}_{i}\) & \(\mathbf{A}_{i}\) & \(\mathbf{F}(\mathbf{t}_{i})\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: The results obtained by the improved average rank method. Using linear correlation coefficients for model fit testing, the formula for calculating the linear correlation coefficient is:  
\[\hat{\rho}=\frac{\sum_{i=1}^{n}x_{i}y_{i}-n\pi\overline{y}}{\sqrt{\left(\sum_{i=1 }^{n}x_{i}^{2}-n\overline{x^{2}}\right)\left(\sum_{i=1}^{n}y_{i}^{2}-n\overline {y^{2}}\right)}} \tag{43}\]  
The linear correlation coefficient using the average rank method \(\hat{\rho_{1}}\) is 0.958, and the linear correlation coefficient using the improved average rank method \(\hat{\rho_{2}}\) is 0.975. When the significance factor \(\alpha=0.01\), the critical value of the linear correlation coefficient \(\hat{\rho_{0}}\) is 0.575. \(\hat{\rho_{2}}>\hat{\rho_{1}}>\hat{\rho_{0}}\); therefore, it is concluded that the effect of linear regression is significant, and using the improved average rank method for regression fitting has better results than using the average rank method. We also use the K-S test method to examine the goodness of fit. The calculation method for the test statistic \(D_{n}\) is as follows:  
\[D_{n}=\sup_{-\infty<x<+\infty}|F_{n}(x)-F_{0}(x)|=\max\{d_{i}\} \tag{44}\]  
where  
\[d_{i}=|F_{n}(t_{i})-F_{0}(t_{i})| \tag{45}\]  
\(F_{n}(x)\) is the empirical distribution function, and \(F_{0}(x)\) is the function value under the assumed distribution. When \(D_{n}\leq D_{n,\alpha}\), the original hypothesis is accepted and it is assumed that it follows the assumed distribution. When significance level \(\alpha=0.1\), \(D_{n,\alpha}=0.272\). When using the average rank method, \(D_{n1}=0.091\). When using the improved average rank method, \(D_{n2}=0.056\).