## 6 Real Data Applications
### Head and Neck Cancer, Dataset 2  
The NGLog-Weib distribution was analysed on the head and neck cancer data and compared with the other distributions to evaluate its optimum fitting power. The summary measures were: min = 12.2, 1 st quartile = 67.210, mean = 223.50, median = 128.50, 3 rd quartile = 219.0, max = 177.60, skewness = 3.384, kurtosis = 16.559, var = 93,286.41, and range = 1763.80. Some basic plots, such as the histogram, kernel density, TTT plot, violin plot, and box plot, for dataset 2 are presented in Figure 6. Similarly, the corresponding parameter values, i.e., \(\hat{\beta}_{MLE}\), \(\hat{\alpha}_{MLE}\), \(\hat{\delta}_{MLE}\), \(\hat{a}_{MLE}\), \(\hat{b}_{MLE}\), and \(\hat{\theta}_{MLE}\) of the NGLog-Weib distribution in comparison with the other competing probability distributions are given in Table 9, whereas the GCMs with \(p\)-values and DMs for selecting the best distribution of the competing probability distributions are presented in Tables 10 and 11. Furthermore, the graphical representation, the estimated PDF, CDF, SF, PP, and QQ plots of the NGLog-Weib distribution, are shown in Figure 7.  
The results presented in Tables 10 and 11 show that the NGLog-Weib distribution has the lowest GFM and DM values and the highest \(p\)-value, as compared to the GFM and DM values and \(p\)-values of the other distributions. For the proposed NGLog-Weib distribution, the AMs and \(p\)-value are: CM = 0.018, AD = 0.119, KS = 0.057, AIC = 560.732, BIC = 566.085, CAIC = 561.332, HQIC = 562.717, and \(p\)-value = 0.9973, respectively. In terms of the CM, AD, KS, AIC, CAIC, HQIC, and \(p\)-value, the second optimum distribution is  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **CM** & **AD** & **KS** & \(p\)**-Values** \\ \hline NGLog–[PERSON] & 0.020 & 0.128 & 0.062 & 0.992 \\ \hline APTra–[PERSON] & 0.093 & 0.554 & 0.106 & 0.672 \\ \hline NExpo–[PERSON] & 0.087 & 0.515 & 0.101 & 0.727 \\ \hline MO–[PERSON] & 0.095 & 0.562 & 0.113 & 0.593 \\ \hline FRLog–[PERSON] & 0.191 & 1.096 & 0.134 & 0.379 \\ \hline Kumar–[PERSON] & 0.022 & 0.136 & 0.065 & 0.985 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The GCMs and \(p\)-values of the competing models for head and neck cancer data (dataset 2).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Models** & **AIC** & **BIC** & **CAIC** & **HQIC** \\ \hline NGLog–Weib & 560.793 & 566.146 & 561.393 & 562.778 \\ \hline APTra–Weib & 567.772 & 573.124 & 568.371 & 569.757 \\ \hline NExpo–Weib & 565.157 & 568.725 & 565.450 & 566.480 \\ \hline MO–Weib & 568.208 & 573.562 & 568.808 & 570.194 \\ \hline FRLog–Weib & 572.884 & 578.236 & 573.484 & 574.869 \\ \hline [PERSON]–Weib & 562.711 & 569.848 & 563.737 & 565.358 \\ \hline \hline \end{tabular}
\end{table}
Table 11: The DMs of the competing models for head and neck cancer, dataset 2.  
Figure 7: Estimated PDF, CDF, HF, CHF, SF, PP plot and QQ plot of the NGLog–Weib distribution for dataset 2.  
the Kumar-Weib model. For the Kumar-Weib model, the values of the CM, AD, KS, AIC, CAIC, HQIC, and \(p\)-value are given by 0.022, 0.136, 0.065, 562.711, 563.737, 565.358 and 0.985, respectively. Similarly, in terms of the BIC, the second optimum distribution is the NExpo-Weib distribution because its value is 568.725.  
In light of the numerical justification in Tables 10 and 11, the visual graphical illustration (the close-fitting of the proposed NGLog-Weib distribution) in Figure 7, and the above discussion, we can observe that the proposed NGLog-Weib distribution is a moral competitor among the other compared probability distributions for the head and neck cancer data (Data 2).