## 6 Misspecification Cases  
For the simulation study, we provided 16 scenarios to validate our proposed algorithm. In this section, we provide the opposite--i.e., we intentionally generate data that are not derived from the Weibull distribution and then analyze them using our proposed algorithm. We generate two datasets taken from different distributions. The first dataset is taken from a double-exponential distribution with location and scale parameters of 0 and 1, respectively, and the second dataset is taken from a logistic distribution with location-scale parameters of 2 and 0.4, respectively. Each of these datasets has as many as 1000 data points from the distribution.  
We used the EV-I mixture and Gaussian mixture distributions to analyze the data described above. We used the same hyperparameters as in the simulation study section because the double-exponential and logistic distributions both have location and scale parameters. To compare the performance between the EV-I mixture and Gaussian mixture distributions, we used the Kullback-Leibler divergence; a complete explanation and formula for the Kullback-Leibler divergence (KLD) can be found in [PERSON] and Harremos [73]. In the simple case, the KLD value of 0 indicates that the true values with fitted densities have identical quantities of information. Thus, the smaller the KLD value, the more identical the true and fitted densities are.  
The posterior distribution of \(k\) for the misspecification cases data can be seen in Table 4. Based on Table 4, the data that we generate are detected with multimodal data, even though the data we generate are unimodal. Then, in Table 5 can be seen the comparison of KLD for EV-I mixture distribution and Gaussian mixture distribution. Based on Table 5, it can be concluded that the EV-I mixture distribution covers more than the Gaussian mixture distribution for these data.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Distribution of Random Data** & \(\boldsymbol{n}\) & \multicolumn{4}{c}{**Pr(\(k\,\boldsymbol{1}\,\boldsymbol{y}\))**} \\ \hline Double-exponential (0,1) & 1000 & \(\Pr(1)=0\) & \(\Pr(2)=0.2519\) & \(\Pr(3)=0.2631\) & \(\Pr(4)=0.1703\) & \(\Pr(5)=0.1182\) \\  & & \(\Pr(6)=0.0789\) & \(\Pr(7)=0.0515\) & \(\sum_{\geq 8}\Pr(k|y)=0.0661\) & \\ \hline Logistic (2,0.4) & 1000 & \(\Pr(1)=0.1492\) & \(\Pr(2)=0.4960\) & \(\Pr(3)=0.2175\) & \(\Pr(4)=0.0843\) & \(\Pr(5)=0.0318\) \\  & & \(\Pr(6)=0.0122\) & \(\Pr(7)=0.0055\) & \(\sum_{\geq 10}\Pr(k|y)=0.0035\) & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Posterior distribution of \(k\) for misspecification cases data based on mixture model using the EV-I distribution.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distribution of** & \multicolumn{2}{c}{**Kullback–Leibler**} \\
**Random Data** & **Number of Components** & \multicolumn{2}{c}{**EV-I Mixture Distribution**} & \multicolumn{2}{c}{**Gaussian Mixture Distribution**} \\ \hline \hline \multirow{6}{*}{Double-exponential (0,1)} & 2 & 0.0935 & 0.5311 \\ \cline{2-4}  & 3 & 0.0789 & 0.5924 \\ \cline{2-4}  & 4 & 0.0902 & 0.6840 \\ \cline{2-4}  & 5 & 0.0873 & 0.3892 \\ \cline{2-4}  & 6 & 0.0789 & 0.2908 \\ \cline{2-4}  & 7 & 0.0677 & 0.2471 \\ \hline \multirow{6}{*}{Logistic (2,0.4)} & 2 & 0.0629 & 0.4920 \\ \cline{2-4}  & 3 & 0.0632 & 0.3288 \\ \cline{1-1} \cline{2-4}  & 4 & 0.0760 & 0.2981 \\ \cline{1-1} \cline{2-4}  & 5 & 0.0754 & 0.2765 \\ \cline{1-1} \cline{2-4}  & 6 & 0.0707 & 0.2587 \\ \cline{1-1} \cline{2-4}  & 7 & 0.0633 & 0.2563 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of Kullback–Leibler divergence using EV-I mixture and Gaussian mixture distributions for misspecification cases data.