## 7 Real Data Applications
### Analysis of First Dataset
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & **Estimates** & **SE** & **KSD** & **PVK** & **AIVC** & **BIVC** & **CAIVC** & **HQIVC** & **CMV** & **ADV** \\ \hline \multirow{3}{*}{UEHLD} & \(\varphi\) & 2.6743 & 0.3089 & \multirow{3}{*}{0.0519} & \multirow{3}{*}{0.0937} & \multirow{3}{*}{\(-25.0405\)} & \multirow{3}{*}{\(-20.8187\)} & \multirow{3}{*}{\(-24.8336\)} & \multirow{3}{*}{\(-23.3859\)} & \multirow{3}{*}{0.0391} & \multirow{3}{*}{0.3333} \\ \cline{2-2} \cline{6-11}  & \(\delta\) & 2.0209 & & & & & & & \\ \hline \multirow{3}{*}{UW} & \(\alpha\) & 1.3395 & 0.1725 & \multirow{3}{*}{0.0682} & \multirow{3}{*}{0.0682} & \multirow{3}{*}{0.09208} & \multirow{3}{*}{\(-24.4872\)} & \multirow{3}{*}{\(-20.2654\)} & \multirow{3}{*}{\(-24.2803\)} & \multirow{3}{*}{\(-22.8326\)} & \multirow{3}{*}{0.0630} & \multirow{3}{*}{0.5097} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 1.7346 & 0.1695 & & & & & & & \\ \hline \multirow{3}{*}{K} & a & 2.3297 & 0.3055 & \multirow{3}{*}{0.0690} & \multirow{3}{*}{0.0941} & \multirow{3}{*}{\(-23.2503\)} & \multirow{3}{*}{\(-19.0285\)} & \multirow{3}{*}{\(-23.0434\)} & \multirow{3}{*}{\(-21.5957\)} & \multirow{3}{*}{0.0527} & \multirow{3}{*}{0.0405} \\ \cline{2-2} \cline{6-11}  & b & 2.7629 & 0.5550 & & & & & & & \\ \hline \multirow{3}{*}{Beta} & \(\alpha\) & 2.7944 & 0.4881 & \multirow{3}{*}{0.0618} & \multirow{3}{*}{0.0629} & \multirow{3}{*}{\(-23.9121\)} & \multirow{3}{*}{\(-19.6903\)} & \multirow{3}{*}{\(-23.7052\)} & \multirow{3}{*}{\(-22.2576\)} & \multirow{3}{*}{0.0491} & \multirow{3}{*}{0.3864} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 2.6041 & 0.4519 & & & & & & & \\ \hline \multirow{3}{*}{KK} & a & 4.6765 & 11.1153 & \multirow{3}{*}{0.0561} & \multirow{3}{*}{0.09850} & \multirow{3}{*}{\(-20.7855\)} & \multirow{3}{*}{\(-12.3420\)} & \multirow{3}{*}{\(-20.0712\)} & \multirow{3}{*}{\(-17.4764\)} & \multirow{3}{*}{0.0484} & \multirow{3}{*}{0.0416} \\ \cline{2-2} \cline{6-11}  & b & 0.8085 & 1.6378 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\alpha\) & 2.4986 & 3.7154 & & & & & & & \\ \hline \multirow{3}{*}{MOK} & \(\beta\) & 0.8259 & 1.2331 & \multirow{3}{*}{0.0582} & \multirow{3}{*}{0.09783} & \multirow{3}{*}{\(-22.6367\)} & \multirow{3}{*}{\(-16.3040\)} & \multirow{3}{*}{\(-22.2156\)} & \multirow{3}{*}{\(-20.1549\)} & \multirow{3}{*}{0.0490} & \multirow{3}{*}{0.4139} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 3.0589 & 0.6447 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\theta\) & 1.9501 & 0.9516 & & & & & & & \\ \hline \multirow{3}{*}{UG} & \(\beta\) & 0.6161 & 0.2661 & \multirow{3}{*}{0.1098} & \multirow{3}{*}{0.4234} & \multirow{3}{*}{\(-17.7518\)} & \multirow{3}{*}{\(-13.5300\)} & \multirow{3}{*}{\(-17.5449\)} & \multirow{3}{*}{\(-16.0972\)} & \multirow{3}{*}{0.1585} & \multirow{3}{*}{1.1540} \\ \cline{2-2} \cline{6-11}  & \(\theta\) & 1.0922 & 0.2472 & & & & & & & \\ \hline \multirow{3}{*}{UGLBXII} & \(\alpha\) & 5.1555 & 8.0816 & \multirow{3}{*}{0.0548} & \multirow{3}{*}{0.9884} & \multirow{3}{*}{\(-22.9406\)} & \multirow{3}{*}{\(-16.6080\)} & \multirow{3}{*}{\(-22.5196\)} & \multirow{3}{*}{\(-20.4588\)} & \multirow{3}{*}{0.0411} & \multirow{3}{*}{0.3522} \\ \cline{2-2} \cline{6-11}  & \(\beta\) & 0.9724 & 0.1862 & & & & & & & \\ \cline{2-2} \cline{6-11}  & \(\lambda\) & 1.8164 & 1.9159 & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: MLE, SE, and measures of models for trade share data. Figure 5: Plots of the estimated PDFs, CDFs, and P-P of the UEHLD of trade share data. Figure 6 sketches the profile log-likelihood of the UEHLD for each parameter by fixing one parameter and varying the other. The figures show that the trade share dataset behaves very well, as we can see that the two roots of the parameters are global maxima. Figure 7 gives the contour plot with varying parameters and log-likelihoods of the UEHLD to confirm the estimates have unique points. Figure 8 shows the trace plots of the posterior distributions of the parameters to track the convergence of the MCMC outputs.