## 6 Real Data Applications
### Injected Guinea Pigs, Dataset 1  
The NGLog-Weib distribution was applied to the injected guinea pigs dataset, and its goodness of fit was compared with the other distributions. The summary measures were: \(\min=10.00\), \(1\)st quartile = \(108.00\), mean = \(176.80\), median = \(149.50\), \(3\)rd quartile = \(224.00\), \(\max=555.00\), skewness = \(1.341\), kurtosis = \(4.989\), variance = \(10\),\(705.10\), and range = \(545.0\). Corresponding to Data 1, the histogram plot, kernel density, TTT plot, violin plot, and box plot are plotted in Figure 4. Similarly, the corresponding model's parameter values, i.e., \(\hat{\beta}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{\delta}_{MLE}\), and \(\hat{\delta}_{MLE}\) of the NGLog-Weib distribution and another extension of the Weib distributions, are recorded in Table 6, whereas the GFMs with \(p\)-values and DMs of the competing probability distributions are shown in Tables 7 and 8. Furthermore, in favour of the AM results presented in Tables 7 and 8, the estimated PDF, CDF, SF, PP (probability-probability), and QQ (quintile-quintile) plots of the NGLog-Weib distribution are plotted in Figure 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & **CM** & **AD** & **KS** & **\(p\)-Values** \\ \hline NGLog-Weib & 0.079 & 0.092 & 0.086 & 0.665 \\ \hline APTra–Weib & 0.804 & 0.138 & 0.171 & 0.029 \\ \hline NExpo–Weib & 0.089 & 0.522 & 0.295 & 0.067 \\ \hline MO–Weib & 0.161 & 0.941 & 0.173 & 0.027 \\ \hline FRLog–[PERSON] & 0.192 & 1.124 & 0.182 & 0.017 \\ \hline Kumar–[PERSON] & 0.082 & 0.523 & 0.092 & 0.580 \\ \hline \hline \end{tabular}
\end{table}
Table 7: The GFMs and \(p\)-values of the competing models for infected guinea pigs data (dataset 1).  
Given the results presented in Tables 7 and 8, we can conclude that the NGLog-Weib distribution has the lowest GFM and DM values and the highest \(p\)-value, as compared to the other distributions. For the proposed NGLog-Weib distribution, the AMs are: CM = 0.079, AD = 0.492, KS = 0.086, AIC = 856.654, BIC = 863.484, CAIC = 857.007, HQIC = 859.373, and \(p\)-value = 0.665. In terms of the AD, KS, AIC, BIC, CAIC, HQIC, and \(p\)-value, the second cohesive distribution is the four-parameter Kumar-Weib model. For the Kumar-Weib model, the AM values of the KS, AIC, BIC, CAIC, HQIC, and \(p\)-value are given by 0.092, 859.492, 868.599, 860.089, 863.117, and 0.580, respectively, while in comparison to the AD, the second-best distribution is the three-parameter APTra-Weib model because for the APTra-Waib model, the value of the statistics, AD, is 0.138.  
Hence, from the numerical depiction in Tables 7 and 8, the visual graphical illustration (the close-fitting of the NGLog-Weib model), and the above comprehensive discussion, we can say that the newly derived NGLog-Weib model is a good competitor among the other compared probability models for the injected guinea pigs dataset.  
Figure 5: Estimated PDF, CDF, HF, CHF, SF, PP plot, and QQ plot of the NGLog–Weib distribution for dataset 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **AIC** & **BIC** & **CAIC** & **HQIC** \\ \hline NGLog–[PERSON] & 856.654 & 863.485 & 857.007 & 859.373 \\ \hline APTra–[PERSON] & 864.127 & 870.957 & 864.479 & 866.846 \\ \hline NExpo–Weib & 902.709 & 907.262 & 902.883 & 904.521 \\ \hline MO–[PERSON] & 865.711 & 872.541 & 866.064 & 868.430 \\ \hline FRLog–[PERSON] & 868.085 & 874.915 & 868.438 & 870.804 \\ \hline Kumar–[PERSON] & 859.492 & 868.599 & 860.089 & 863.117 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The DMs of the competing models for infected guinea pigs data (dataset 1).