## 6 Performance Analysis by Monte Carlo Simulation
### Simulation Study  
First, we set the true value with various parameter values of the UEHLD as: \(\varphi=4\) and \(\delta=0.5\), \(2\), \(4\) in Table 3, while \(\varphi=0.5\), \(2\) and \(\delta=0.5\), \(2\), \(4\) in Table 4. Altogether, nine sets of simulations of the UEHLD data with different sample sizes as \(n=30\), \(75\), and \(150\) were generated. To avoid the starting bias, MSE, and length of CIs, 10,000 points were generated for each sample simulation. The generated data of the UEHLD were obtained by using QF (4). The estimates of the ML, MPS, and Bayesian techniques were obtained, and we used the NR algorithm for the numerical analysis to obtain the ML estimates (MLEs) and MPS estimates, as well as the MH algorithm to obtain the Bayesian estimates. The iterative algorithms were used to obtain 10,000 estimates for each parameter of the UEHLD when the first initial one was the actual parameter. In CI, we used the 5% level of significance. This simulation study was implemented via R packages. Simulation algorithm: By creating all simulation controls, we may develop our model. The steps below must be completed in this stage in the following order:  
Step 1: Assume various values for the sample size, as well as the UEHLD parameter vector. Step 2: Using the QF, create the sample random values for the UEHLD. Step 3: To acquire the estimators of the parameters for the UEHLD, we computed by solving non-linear equations for each estimate technique. Step 4: Perform this experiment (\(\text{L}-1\)) times. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & & & & **ML** & & & **MPS** & & & **Bayesian** \\ \hline \(\varphi\) & \(\delta\) & \(n\) & & **Bias** & **MSE** & **LACI** & **Bias** & **MSE** & **LACI** & **Bias** & **MSE** & **LCCI** \\ \hline \multirow{8}{*}{0.5} & \multirow{4}{*}{30} & & 2.0073 & 5.2698 & 4.3686 & 1.1855 & 2.0800 & 3.2213 & 0.0099 & 0.0206 & 0.5464 \\ \cline{3-11}  & & & \(\delta\) & 0.2168 & 0.0690 & 0.5818 & 0.1290 & 0.0308 & 0.4674 & 0.0349 & 0.0074 & 0.2975 \\ \cline{3-11}  & & & \(\varphi\) & 1.7336 & 3.3980 & 2.4577 & 1.4091 & 2.2988 & 2.1950 & 0.0082 & 0.0793 & 0.3429 \\ \cline{3-11}  & & & \(\delta\) & 0.1835 & 0.0402 & 0.3161 & 0.1474 & 0.0270 & 0.2856 & 0.0310 & 0.0342 & 0.1868 \\ \cline{3-11}  & & & \(\varphi\) & 1.7281 & 3.2058 & 1.8377 & 1.0685 & 1.9471 & 1.6054 & 0.0078 & 0.0298 & 0.2079 \\ \cline{3-11}  & & & \(\delta\) & 0.1817 & 0.0358 & 0.2078 & 0.1180 & 0.0312 & 0.1958 & 0.0235 & 0.0169 & 0.1289 \\ \hline \multirow{8}{*}{4} & \multirow{4}{*}{2} & & \(\varphi\) & 1.3661 & 2.3880 & 2.8328 & 0.8318 & 1.0361 & 2.3012 & 0.0060 & 0.0204 & 0.5552 \\ \cline{3-11}  & & & \(\delta\) & 1.7326 & 4.2751 & 4.4255 & 0.9446 & 1.3462 & 2.6423 & 0.0116 & 0.0187 & 0.5409 \\ \cline{3-11}  & & & \(\varphi\) & 1.2307 & 1.7054 & 1.7132 & 0.9823 & 1.1135 & 1.5118 & 0.0042 & 0.0740 & 0.3301 \\ \cline{3-11}  & & & \(\delta\) & 1.3928 & 2.2318 & 2.1193 & 1.0534 & 1.2833 & 1.6344 & 0.0124 & 0.0796 & 0.3451 \\ \cline{3-11}  & & & \(\varphi\) & 1.2105 & 1.5649 & 1.2379 & 0.7713 & 1.0360 & 1.1219 & 0.0044 & 0.0290 & 0.2062 \\ \cline{3-11}  & & & \(\delta\) & 1.3343 & 1.9335 & 1.5348 & 0.8278 & 1.2356 & 1.2745 & 0.0075 & 0.0314 & 0.2189 \\ \hline \multirow{8}{*}{4} & \multirow{4}{*}{30} & & \(\varphi\) & 1.2142 & 1.8066 & 2.2608 & 0.5266 & 0.4207 & 1.4846 & 0.0025 & 0.0191 & 0.5404 \\ \cline{3-11}  & & & \(\delta\) & 2.3127 & 8.8163 & 8.9583 & 1.4443 & 2.6906 & 3.0498 & 0.0076 & 0.0211 & 0.5526 \\ \cline{1-1} \cline{3-11}  & & & \(\varphi\) & 1.1564 & 1.4955 & 1.5608 & 0.6366 & 0.4640 & 0.9504 & 0.0051 & 0.0770 & 0.3365 \\ \cline{1-1} \cline{3-11}  & & & \(\delta\) & 2.0123 & 7.7098 & 6.3376 & 1.8318 & 3.6308 & 2.0584 & 0.0075 & 0.0837 & 0.3514 \\ \cline{1-1} \cline{3-11}  & & & \(\varphi\) & 1.1326 & 1.3535 & 1.0424 & 0.4978 & 0.4368 & 0.6574 & 0.0012 & 0.0283 & 0.2052 \\ \cline{1-1} \cline{3-11}  & & & \(\delta\) & 1.7292 & 6.9873 & 3.8550 & 1.3834 & 2.7085 & 1.3902 & 0.0032 & 0.0321 & 0.2203 \\ \hline \end{tabular}
\end{table}