## 7 Real Data Applications
### Analysis of Second Dataset  
**First**, we compared the fit of the proposed UEHLD with that of the beta, K, KK, MOK, UB, UBXII, and UGLBXII distributions. We used the aforementioned criterion measurements found in Table 8 to compare the fit models. Table 8 reveals that the UEHLD has smaller measures than the other competing distributions, indicating that it offers a superior fit. The estimated PDF, empirical CDF, and P-P plots for Dataset II are shown in Figure 9. These graphical plots support the outcomes in Table 9.  
Figure 8: MCMC trace and posterior distribution for the UEHLD parameters by trade share data. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & **Estimates** & **SE** & **KSD** & **PKS** & **AIVC** & **BIVC** & **CAIVC** & **HQIVC** & **CMV** & **ADV** \\ \hline \multirow{2}{*}{UEHLD} & \(\varphi\) & 1.1281 & 0.2051 & \multirow{2}{*}{0.0565} & \multirow{2}{*}{0.9999} & \multirow{2}{*}{\(-\)3.1043} & \multirow{2}{*}{\(-\)0.3019} & \multirow{2}{*}{\(-\)2.6599} & \multirow{2}{*}{\(-\)2.2078} & \multirow{2}{*}{0.0150} & \multirow{2}{*}{0.1158} \\ \cline{2-2} \cline{6-11}  & \(\delta\) & 1.2346 & 0.2951 & & & & & & & \\ \hline \multirow{2}{*}{K} & a & 0.9627 & 0.2017 & \multirow{2}{*}{0.0650} & \multirow{2}{*}{0.9987} & \multirow{2}{*}{\(-\)2.6221} & \multirow{2}{*}{0.1803} & \multirow{2}{*}{\(-\)2.1776} & \multirow{2}{*}{\(-\)1.7256} & \multirow{2}{*}{0.0183} & \multirow{2}{*}{0.1551} \\ \cline{2-2} \cline{6-11}  & b & 1.6084 & 0.4137 & & & & & & & \\ \hline \multirow{2}{*}{Beta} & \(\alpha\) & 0.9666 & 0.2238 & \multirow{2}{*}{0.0669} & \multirow{2}{*}{0.9979} & \multirow{2}{*}{\(-\)2.6101} & \multirow{2}{*}{0.1923} & \multirow{2}{*}{\(-\)2.1657} & \multirow{2}{*}{\(-\)1.7136} & \multirow{2}{*}{0.0184} & \multirow{2}{*}{0.1559} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 1.6205 & 0.4107 & & & & & & & \\ \hline \multirow{2}{*}{KK} & a & 7.9330 & 1.2565 & \multirow{2}{*}{0.0714} & \multirow{2}{*}{0.0951} & \multirow{2}{*}{\(-\)0.2150} & \multirow{2}{*}{5.3898} & \multirow{2}{*}{1.3850} & \multirow{2}{*}{1.5780} & \multirow{2}{*}{0.0163} & \multirow{2}{*}{0.1289} \\ \cline{2-2} \cline{6-11}  & b & 0.4949 & 0.0478 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\alpha\) & 8.7493 & 0.0614 & & & & & & & \\ \hline \multirow{2}{*}{MOK} & \(\beta\) & 0.1404 & 0.0287 & \multirow{2}{*}{0.0628} & \multirow{2}{*}{0.9992} & \multirow{2}{*}{\(-\)1.2087} & \multirow{2}{*}{2.9949} & \multirow{2}{*}{\(-\)0.2856} & \multirow{2}{*}{0.1361} & \multirow{2}{*}{0.0151} & \multirow{2}{*}{0.1164} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 1.1872 & 0.3472 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\theta\) & 1.2585 & 0.6458 & & & & & & & \\ \hline \multirow{2}{*}{UBXII} & \(\beta\) & 1.0331 & 0.2060 & \multirow{2}{*}{0.0993} & \multirow{2}{*}{0.9008} & \multirow{2}{*}{1.9220} & \multirow{2}{*}{4.7244} & \multirow{2}{*}{2.3665} & \multirow{2}{*}{2.8185} & \multirow{2}{*}{0.0586} & \multirow{2}{*}{0.4419} \\ \cline{2-2} \cline{6-11}  & \(\theta\) & 1.8465 & 0.3054 & & & & & & & \\ \hline \multirow{2}{*}{UGLBXII} & \(\alpha\) & 582.6661 & 45.8072 & \multirow{2}{*}{0.0570} & \multirow{2}{*}{0.9999} & \multirow{2}{*}{\(-\)1.4258} & \multirow{2}{*}{2.7778} & \multirow{2}{*}{\(-\)0.5027} & \multirow{2}{*}{\(-\)0.0811} & \multirow{2}{*}{0.0163} & \multirow{2}{*}{0.1166} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 0.6820 & 0.1076 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\lambda\) & 160.0645 & 25.8435 & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: MLE, SE, and measures of models for polyester fibers’ tensile strength data. **Second**, utilizing the information on the tensile strength of the fibers, we determined the ML, MPS, and Bayesian estimates using the SELF of the UEHLD's parameters, which are listed in Table 9. We used the mentioned MCMC algorithm to generate 10,000 MCMC samples. The initial values of the unknown parameters were taken to be their MLEs in order to use the MCMC sampling procedure.