## 6 Real Data Applications
### Patients Receiving Analgesics, Dataset 4  
Finally, the NGLog-Weib distribution was evaluated on the patients receiving analgesics' datasets. The key measures were: minimum = 1.100, 1 st quartile = 1.475, mean = 1.900, median = 1.700, 3 rd quartile = 2.050, maximum = 4.100, skewness= 1.720, kurtosis= 5.924, var= 0.496, and range = 3.0. Figure 10 depicts the histogram, kernel density, TTT plot, violin plot, and box plot for patients receiving the analgesics, dataset 4. Similarly, the values of the parameters, i.e., \(\hat{\beta}_{MLE}\), \(\hat{\kappa}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{\delta}_{MLE}\), and \(\hat{\theta}_{MLE}\) of the NGLog-Weib, with other well-known competing distributions, are displayed in Table 15, whereas the GFMs with \(p\)-values and DMs for selecting the best distribution of the competing distributions are provided in Tables 16 and 17. Furthermore, the graphical representation of the estimated PDF, CDF, SF, PP, and QQ plots of the NGLog-Weib distribution is shown in Figure 11.  
The results presented in Tables 16 and 17 display that the NGLog-Weib distribution has the least AM values and the highest \(p\)-value, as compared to the AMs and \(p\)-values of the other distributions. For the proposed NGLog-Weib distribution, the AMs and \(p\)-value are: CM = 0.058, AD = 0.344, KS = 0.146, AIC = 38.699, BIC = 41.687, CAIC = 40.199, HQIC = 39.282, and \(p\)-value = 0.850, respectively. In terms of the CM, AD, KS, AIC, HQIC, and \(p\)-value, the second most optimum distribution is the Kumar-Weib model. For the Kumar-Weib model, the values of the CM, AD, KS, AIC, HQIC, and \(p\)-value are 0.060, 0.359, 0.156, 39.924, 40.702, and 0.833, respectively. Similarly, in terms of the BIC, CAIC,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **CM** & **AD** & **KS** & \(p\)**-Values** \\ \hline NGLog–[PERSON] & 0.058 & 0.344 & 0.146 & 0.850 \\ \hline APTra–[PERSON] & 0.132 & 0.779 & 0.158 & 0.701 \\ \hline NExpo–Weib & 0.153 & 0.904 & 0.181 & 0.527 \\ \hline MO–Weib & 0.076 & 0.457 & 0.165 & 0.802 \\ \hline FRLog–Weib & 0.141 & 0.852 & 0.223 & 0.273 \\ \hline Kumar–Weib & 0.060 & 0.359 & 0.156 & 0.833 \\ \hline \hline \end{tabular}
\end{table}
Table 16: The GFMs and \(p\)-values of the competing models for the patients receiving analgesics’ data (dataset 4).  
Figure 10: For dataset 4, the (**a**) histogram, (**b**) kernel density plot, (**c**) TTT plot, (**d**) violin plot, and (**e**) box plot.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **AIC** & **BIC** & **CAIC** & **HQIC** \\ \hline NGLog–[PERSON] & 38.699 & 41.687 & 40.199 & 39.282 \\ \hline APTra–[PERSON] & 43.418 & 46.404 & 44.917 & 44.000 \\ \hline NExpo–[PERSON] & 42.936 & 44.927 & 43.642 & 43.325 \\ \hline MO–[PERSON] & 40.781 & 43.769 & 42.283 & 41.363 \\ \hline FRLog–[PERSON] & 44.704 & 47.691 & 46.203 & 45.286 \\ \hline Kumar–[PERSON] & 39.924 & 43.907 & 42.591 & 40.702 \\ \hline \hline \end{tabular}
\end{table}
Table 17: The DMs of the competing models for the patients receiving analgesics’ data (dataset 4).  
and HQIC, the second-best distribution is the MO-Weib model because its BIC, CAIC, and HQIC values are 43.768, 42.281, and 41.364, respectively.  
Based on the justification of the numerical description in Tables 16 and 17, the visual graphical illustration, and the above comprehensive discussion, we can observe that the proposed NGLog-Weib model is a good (or optimum) competitor among the other competing distributions for the patients receiving analgesics (dataset 4).