## 4 Numerical Examples
### Results  
#### 4.2.1 Model Analysis  
Dataset #1, presented in Table 3, was reported by [31]. The failure data come from two releases of a large medical record system (LMRS). The week index is from 1 week to 18 weeks, and there are 176 cumulative failures at 18 weeks in Dataset #1. In Dataset #2, the week index is from 1 week to 17 weeks, and there are 204 cumulative failures at 17 weeks. Detailed information can be seen in [31]. Dataset #3, given in Table 4, was reported by [32] based on system test data for a telecommunication system (TS data set). In Dataset #3, the week index is from 1 week to 21 weeks, and there are 43 cumulative failures at 21 weeks. Detailed information can be seen in [32]. \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Criteria** & **Formula** \\ \hline
1 & MSE & \(\frac{\sum_{\mathrm{i}=0}^{\mathrm{i}}(\mathrm{\hat{m}(t)}-\mathrm{y_{i}})^{2}}{ \mathrm{n}-\mathrm{m}}\) \\
2 & RMSE & \(\sqrt{\frac{\sum_{\mathrm{i}=0}^{\mathrm{i}}(\mathrm{\hat{m}(t)}-\mathrm{y_{i}}) ^{2}}{\mathrm{n}-\mathrm{m}}}\) \\
3 & AIC [29] & \(-2\log\mathrm{L}+2\mathrm{m}\) \\
4 & \(\mathrm{R^{2}}\)[8] & \(1-\frac{\sum_{\mathrm{i}=0}^{\mathrm{i}}(\mathrm{\hat{m}(t)}-\mathrm{y_{i}})^{2} }{\sum_{\mathrm{i}=0}^{\mathrm{i}}(\mathrm{y_{i}}-\mathrm{y_{i}})^{2}}\) \\
5 & \(\mathrm{Adj~{}R^{2}}\)[8] & \(1-\frac{\mathrm{(1-R^{2})}(\mathrm{n}-\mathrm{m})}{\mathrm{n}-\mathrm{m}}\) \\
6 & SAE [10] & \(\sum_{\mathrm{i}=0}^{\mathrm{i}}|\mathrm{\hat{m}(t)}-\mathrm{y_{i}}|\) \\
7 & PRR [30] & \(\frac{\sum_{\mathrm{i}=0}^{\mathrm{i}}\left(\frac{\mathrm{\hat{m}(t)}-\mathrm{y_{ i}}}{\mathrm{\hat{m}(t)}}\right)^{2}}{\mathrm{\hat{m}(t)}}\) \\
8 & PP [30] & \(\frac{\sum_{\mathrm{i}=0}^{\mathrm{i}}\left(\frac{\mathrm{\hat{m}(t)}-\mathrm{y_{ i}}}{\mathrm{y_{i}}}\right)^{2}}{\mathrm{y_{i}}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of criteria for model comparisons. MSE (mean squared error), RMSE (root mean squared error), AIC ([PERSON]’s information criterion), \(\mathrm{R^{2}}\) (correlation index of the regression curve equation), \(\mathrm{Adj~{}R^{2}}\) (adjicted \(\mathrm{R^{2}}\)), SAE (sum of absolute error), PRR (predictive ratio risk), and PP (predictive power)We obtained the estimated parameters and the eight common criteria in Table 2 of all 13 models at \(t=1,2,3,\cdots,\) 18 from Dataset #1, at \(t=1,2,3,\cdots,\) 17 from Dataset #2, and at \(t=1,\ 2,\ 3,\ \cdots,\) 21 from Dataset #3. We used the LSE method for parameter estimation. Table 5 shows the estimated parameters and Tables 6-8 show the values of the eight common criteria for all 13 models. The closer the values of MSE, RMSE, AIC, SAE, PRR, and PP are to 0 and the closer the values of R\({}^{2}\) and Adj R\({}^{2}\) are to 1, the better. As a result, Table 6 shows that for the proposed model, the values of MSE, RMSE, AIC, SAE, and PP are 93.2910, 9.6587, 182.2178, 99.6157, and 0.9340, respectively, which are lower than those of the other models. In addition, the values of R\({}^{2}\) and Adj R\({}^{2}\) are 0.9836 and 0.9726, respectively, \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Week Index**} & \multicolumn{2}{c}{**Exposure Time**} & \multirow{2}{*}{**Failures**} & \multirow{2}{*}{**Cumulative Failures**} \\  & & & \\  & **(Cumulative System Test Hours)** & & \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: System test data for a telecommunication system (TS)—Dataset #3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Week Index**} & \multicolumn{2}{c}{**Dataset \#1**} & \multicolumn{2}{c}{**Dataset \#2**} \\ \cline{2-5}  & **Failures** & **Cumulative Failures** & **Failures** & **Cumulative Failures** \\ \hline  
\\ \hline \hline \end{tabular}
\end{table} 
Table 3: Failure data of large medical record system (LMRS)—Dataset #1 and #2. Figure 11: Comparison of RMSE (root mean squared error) and AIC (Akaike’s information criterion) for Dataset #1. Figure 12: Comparison of RMSE and AIC for Dataset #2.  
reliability is 0.8641.