## 6 Application to Real Data
### Second Reactor Pump Data  
We utilized a data set that describes the time between failure of secondary reactor pumps [31]. The data set is illustrated in Table 2. It was used by previous researchers [6], and it is considered to be a data set that has a upside-down bathtub-shaped hazard rate. The TTT plot and the box plot are presented in Figure 7. The box plot suggests that the data are heavily right-skewed. The TTT plot suggests that the data are associated with a UBT hazard function.  
Table 3 presented the estimates with corresponding standard errors, LRT test statistic, Wald test statistic, and corresponding \(p\)-values from both tests. We were able to reject the null hypothesis for both of the tests at the significance level \(\alpha=0.05\). Thus, we concluded that EEP is more proper compared to EP when fitting to the data set. The KS test also suggested that we could not reject the null hypothesis of the data being from an EEP distribution.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline \multirow{2}{*}{\(\theta\)} & **Sample** & \multicolumn{2}{c}{**LRT**} & \multicolumn{2}{c}{**Wald’s Test**} \\ \cline{3-6}  & **Size (\(n\))** & \(\mathbf{\alpha=0.01}\) & \(\mathbf{\alpha=0.05}\) & \(\mathbf{\alpha=0.01}\) & \(\mathbf{\alpha=0.05}\) \\ \hline \multirow{4}{*}{0.01} & \(n=50\) & 0.0110 & 0.0536 & 0.0107 & 0.0484 \\ \cline{2-6}  & \(n=100\) & 0.0125 & 0.0496 & 0.0105 & 0.0477 \\ \cline{2-6}  & \(n=200\) & 0.0095 & 0.0499 & 0.0101 & 0.0503 \\ \hline \multirow{4}{*}{0.1} & \(n=50\) & 0.0094 & 0.0514 & 0.0087 & 0.0443 \\ \cline{2-6}  & \(n=100\) & 0.0108 & 0.0529 & 0.0104 & 0.0519 \\ \cline{2-6}  & \(n=200\) & 0.0112 & 0.0495 & 0.0113 & 0.0517 \\ \hline \multirow{4}{*}{1} & \(n=50\) & 0.0116 & 0.0537 & 0.0111 & 0.0469 \\ \cline{2-6}  & \(n=100\) & 0.0120 & 0.0542 & 0.0121 & 0.0523 \\ \cline{2-6}  & \(n=200\) & 0.0100 & 0.0484 & 0.0113 & 0.0508 \\ \hline \multirow{4}{*}{10} & \(n=50\) & 0.0120 & 0.0544 & 0.0095 & 0.0490 \\ \cline{2-6}  & \(n=100\) & 0.0104 & 0.0523 & 0.0103 & 0.0495 \\ \cline{1-1} \cline{2-6}  & \(n=200\) & 0.0121 & 0.0518 & 0.0127 & 0.0523 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Type 1 error rates for likelihood ratio test and asymptotic Wald’s test.  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline  & 2.160, 0.150, 4.082, 0.746, 0.358, 0.199, 0.402, 0.101, 0.605, 0.954, \\  & 1.359, 0.273, 0.491, 3.465, 0.070, 6.560, 1.060, 0.062, 4.992, 0.614, \\  & 5.320, 0.347, 1.921 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Second reactor pump failure time data.  
A comparisons of the four different models is presented in Table 4. In terms of all GoF measures, the EEP model demonstrated better performances compared to the original EP model. Moreover, the EEP model presented comparable performances against the heavy-tailed Pareto model and the Weibull-Pareto model, with less parameters. This essentially suggests that the EEP model is capable of explaining the data features with only two parameters. Figure 8 contains the plot-fitted EEP hazard function and the histogram with the fitted pdf. The fitted hazard function possesses a UBT shape. Figure 9 demonstrates a comparison of the empirical survival function and the fitted EEP survival function. It can be clearly seen that EEP provided great performance when fitting to the second reactor pump data set.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Model** & \(p\) & **AIC** & **AIC3** & **AICc** & **CAIC** \\ \hline EEP & 2 & 72.29528 & 74.29528 & 72.89528 & 76.56627 \\ \hline EP & 1 & 76.62130 & 77.62130 & 76.81178 & 78.75679 \\ \hline Generalized Heavy-Tailed Pareto & 3 & 69.86288 & 72.86288 & 71.12604 & 76.26936 \\ \hline Weighted Weibull–Pareto Composite & 4 & 69.71532 & 73.73368 & 71.95590 & 78.27566 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of the four models in terms of the five GoF measures.  
Figure 7: TTT plot (**left**) and box plot (**right**) of the second reactor pump data set.