## 7 Real Data Applications
### Analysis of COVID-19 Data  
**First**, we compared the fit of the proposed UEHLD with that of the UW, K, KK, MOK, and UG distributions. We used the aforementioned criterion measurements found in Table 10 to compare the fit models. Table 10 reveals that the UEHLD has smaller measures than other competing distributions, indicating that it offers a superior fit for COVID-19 data. We should also point out that, in comparison to the models reported by [PERSON] et al. [37], the findings of our new model yield superior measure values.  
Figure 11: Contour plots of log functions with different UEHLD parameter values of polyester fibers’ tensile data.  
Figure 12: MCMC trace and posterior distribution for the UEHLD parameters using polyester fibers’ tensile data.  
The estimated PDF, empirical CDF, and P-P plots for the COVID-19 data are shown in Figure 13. These graphical plots support the outcomes in Table 10.  
Second, for the COVID-19 data, we determined the ML, MPS, and Bayesian estimates using the SELF of the UEHLD's parameters for COVID-19 data, which are listed in Table 11. 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. For the COVID-19 data, Figure 14 draws the profile log-likelihood of the UEHLD for each parameter by fixing one parameter and changing the others. Figure 15 demonstrates the excellent behavior of the aforementioned data, since the two roots of the parameters are global maxima.  
Figure 16 shows trace plots of the posterior distributions of the parameters to track the convergence of the MCMC outputs. Additionally, they display the histograms for the marginal posterior density estimates of the parameters based on 10,000 chain values and the Gaussian kernel, demonstrating how effectively the MCMC process converges. All  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & **Estimates** & **SE** & **KSD** & **PVKS** & **AICVC** & **BICVC** & **CAICVC** & **HQICVC** & **CVMV** & **ADV** \\ \hline \multirow{3}{*}{UEHLD} & \(\varphi\) & 1.2515 & 0.1030 & & & & & & & & \\ \cline{2-10}  & \(\delta\) & 29.3838 & 9.3197 & & & & & & & & \\ \hline \multirow{3}{*}{UW} & \(\alpha\) & 0.0024 & 0.0003 & & & & & & & & \\ \cline{2-10}  & \(\beta\) & 4.3135 & 0.1105 & & & & & & & & \\ \hline \multirow{3}{*}{K} & a & 1.2399 & 0.1055 & & & & & & & & \\ \cline{2-10}  & b & 55.7476 & 18.3042 & & & & & & & & \\ \hline \multirow{3}{*}{KK} & a & 2.4673 & 0.8418 & & & & & & & & \\ \cline{2-10}  & b & 0.5254 & 0.1184 & & & & & & & & \\ \cline{2-10}  & \(\alpha\) & 3.9566 & 0.9116 & & & & & & & & \\ \cline{2-10}  & \(\beta\) & 3.8605 & 1.1342 & & & & & & & & \\ \hline \multirow{3}{*}{MOK} & a & 0.0119 & 0.0225 & & & & & & & & & \\ \cline{2-10}  & \(\beta\) & 1.3908 & 0.1793 & & & & & & & & \\ \cline{2-10}  & \(\theta\) & 1.8396 & 2.6201 & & & & & & & & \\ \hline \multirow{3}{*}{UG} & \(\beta\) & 0.0180 & 0.0071 & & & & & & & & & \\ \cline{2-10}  & \(\theta\) & 0.9767 & 0.0803 & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 10: MLE, SE, and measures of models for COVID-19 data.  
Figure 13: Plots of the estimated PDFs, CDFs, and P-P of the UEHLD of COVID-19 data.