## 5 Simulation Study  
In this section, we have 16 scenarios, namely Weibull mixture distribution with two components, three components, four components, and five components, each of which is generated with a sample of 125, 250, 500, and 1000 per component. Detailed descriptions of each scenario are given in Table 1, where the \"Parameter of EV-I distribution\" column is transformed from the \"Parameter of Weibull distribution\" column. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & **Number of** & \multirow{2}{*}{**Component**} & \multirow{2}{*}{**Number of** & \multirow{2}{*}{**Parameter of Weibull**} & \multirow{2}{*}{**Parameter of EV-I**} \\  & & & & **Distribution** & & **Distribution** \\ \hline \multirow{2}{*}{1} & \multirow{2}{*}{2} & 1 st & 125 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 125 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \hline \multirow{2}{*}{2} & \multirow{2}{*}{2} & 1 st & 250 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 250 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \hline \multirow{2}{*}{3} & \multirow{2}{*}{2} & 1 st & 500 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 500 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \hline \multirow{2}{*}{4} & \multirow{2}{*}{2} & 1 st & 1000 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 1000 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \hline \multirow{2}{*}{5} & \multirow{2}{*}{3} & 1 st & 125 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 125 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 125 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \hline \multirow{2}{*}{6} & \multirow{2}{*}{3} & 1 st & 250 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 250 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 250 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \hline \multirow{2}{*}{7} & \multirow{2}{*}{3} & 1 st & 500 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 125 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 125 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \cline{3-8}  & & 4 th & 125 & 1/0.2 & \(\exp(5)\) & 5 & 0.2 \\ \hline \multirow{2}{*}{10} & \multirow{2}{*}{4} & 1 st & 250 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 250 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 250 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \cline{3-8}  & & 4 th & 250 & 1/0.2 & \(\exp(5)\) & 5 & 0.2 \\ \hline \multirow{2}{*}{11} & \multirow{2}{*}{4} & 1 st & 500 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 500 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 500 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \cline{3-8}  & & 4 th & 500 & 1/0.2 & \(\exp(5)\) & 5 & 0.2 \\ \hline \multirow{2}{*}{11} & \multirow{2}{*}{4} & 1 st & 500 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 500 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 500 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \cline{3-8}  & & 4 th & 500 & 1/0.2 & \(\exp(5)\) & 5 & 0.2 \\ \hline \multirow{2}{*}{11} & \multirow{2}{*}{4} & 1 st & 500 & 1 & \(\exp(-2)\) & \(-2\) & 1 \\ \cline{3-8}  & & 2 nd & 500 & 1/0.5 & \(\exp(0)\) & 0 & 0.5 \\ \cline{3-8}  & & 3 rd & 500 & 1/1.5 & \(\exp(3)\) & 3 & 1.5 \\ \cline{3-8}  & & 4 th & 500 & 1/0.2 & \(\exp(5)\) & 5 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sixteen scenarios of the Weibull mixture distribution and their transformation into the EV-I mixture distribution. In these scenarios, our specific choices for the hyperparameters were \(\zeta=R\), \(\theta=2\), \(g=0.2\), \(h=\frac{10}{R^{2}}\), \(\delta=1\), and \(k_{\max}=30\), where \(R\) and \(\varepsilon\) are the length and midpoint (median) of the observed data, respectively (see [PERSON] and [PERSON] [16]). Based on the selection of the hyperparameters, we performed an analysis with 200,000 sweeps. With these 200,000 sweeps, we got a value of \(k\) as high as 200,000. From this, we took the most frequently occurring (mode) of \(k\). Then, we replicated this step 500 times. Thus, we already had one mode \(k\) for each replication.