$\hat{\beta}$ & 1.8390 & 2.1037 & 2.6070 & 2.7614 \\

RMSE($\hat{\mu}$) & 0.4589 & 0.3289 & 0.1454 & 0.0960 \\

RMSE($\hat{\sigma}$) & 0.4822 & 0.3451 & 0.1376 & 0.0875 \\

RMSE($\hat{\beta}$) & 1.1776 & 0.9268 & 0.4519 & 0.3024 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

\caption{MLE and RMSE for $\mu=3, \sigma=2, \beta=1$}

\centering

\begin{tabular}{|c|c|c|c|c|}

\hline

$n$ & 50 & 100 & 500 & 1000 \\

\hline

$\hat{\mu}$ & 2.9945 & 2.9978 & 3.0006 & 3.0003 \\

$\hat{\sigma}$ & 1.8414 & 1.9692 & 2.0369 & 2.0413 \\

$\hat{\beta}$ & 1.1178 & 1.0602 & 1.0459 & 1.0465 \\

\text{RMSE}($\hat{\mu}$) & 0.034 & 0.0152 & 0.0022 & 0.001 \\

\text{RMSE}($\hat{\sigma}$) & 0.3739 & 0.2011 & 0.0768 & 0.0701 \\

\text{RMSE}($\hat{\beta}$) & 0.2121 & 0.1248 & 0.0651 & 0.0659 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

\caption{MLE and RMSE for $\mu=3, \sigma=3, \beta=1$}

\centering

\begin{tabular}{|c|c|c|c|c|}

\hline

$n$ & 50 & 100 & 500 & 1000 \\

\hline

$\hat{\mu}$ & 3.0032 & 2.9939 & 3.0006 & 3.0007 \\

$\hat{\sigma}$ & 2.6142 & 2.8153 & 3.0089 & 3.0307 \\

$\hat{\beta}$ & 1.1252 & 1.0795 & 1.0382 & 1.0374 \\

\text{RMSE}($\hat{\mu}$) & 0.0552 & 0.0246 & 0.0078 & 0.0019 \\

\text{RMSE}($\hat{\sigma}$) & 0.6651 & 0.4537 & 0.1328 & 0.062 \\

\text{RMSE}($\hat{\beta}$) & 0.2025 & 0.1591 & 0.0668 & 0.0534 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

\caption{MLE and RMSE for $\mu=2, \sigma=1, \beta=1$}

\centering

\begin{tabular}{|c|c|c|c|c|}

\hline

$n$ & 50 & 100 & 500 & 1000 \\

\hline

$\hat{\mu}$ & 1.9970 & 2.0002 & 2.0001 & 2.0000 \\

$\hat{\sigma}$ & 1.0015 & 1.0358 & 1.0505 & 1.0639 \\

$\hat{\beta}$ & 1.0893 & 1.0561 & 1.0600 & 1.0694 \\

RMSE($\hat{\mu}$) & 0.0194 & 0.0074 & 0.0010 & 0.0004 \\

RMSE($\hat{\sigma}$) & 0.1348 & 0.0869 & 0.0851 & 0.0932 \\

RMSE($\hat{\beta}$) & 0.1748 & 0.0947 & 0.0853 & 0.0935 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

\caption{MLE and RMSE for $\mu=1, \sigma=1, \beta=1$}

\centering

\begin{tabular}{|c|c|c|c|c|}

\hline

$n$ & 50 & 100 & 500 & 1000 \\

\hline

$\hat{\mu}$ & 0.9970 & 1.0003 & 1.0001 & 1.0000 \\

$\hat{\sigma}$ & 1.0007 & 1.0364 & 1.0504 & 1.0639 \\

$\hat{\beta}$ & 1.0892 & 1.0554 & 1.0600 & 1.0694 \\

RMSE($\hat{\mu}$) & 0.0193 & 0.0069 & 0.0010 & 0.0005 \\

RMSE($\hat{\sigma}$) & 0.1349 & 0.0856 & 0.0850 & 0.0931 \\

RMSE($\hat{\beta}$) & 0.1738 & 0.0913 & 0.0853 & 0.0935 \\

\hline

\end{tabular}

\end{table}

\subsection{parative Analysis}

The above data tables provide a detailed presentation of the results from multiple simulations, including estimates $\hat{\mu}$, $\hat{\sigma}$, and $\hat{\beta}$ under different sample sizes n, along with their corresponding RMSE values. The following is a summary and analysis of the results.

\subsubsection{Influence of Sample Size}

From the tables, it can be observed that as the sample size increases, the RMSE of each estimate decreases, indicating an improvement in the uracy of the estimates with increasing sample size. At the same time, the estimates of \(\hat{\mu}\), \(\hat{\sigma}\), and \(\hat{\beta}\) also exhibit a trend of gradual stabilization with increasing sample size.

\begin{figure}[ht!] %!t

\centering

\includegraphics[width=3.5in]{setting1.png}

\caption{The variation trend of each parameter estimators and RMSE with sample size of MOM and MLE in setting1: $\mu$ = 3, $\sigma$ = 1, and $\beta$ = 1}

\label{LP}

\end{figure}

\subsubsection{MOM vs MLE}

Additionally, it also can be observed that there is some difference between the two different methods for a given sample size and set conditions. Especially for small sample sizes (n=50, 100), moment estimates tend to be more unstable than maximum likelihood estimates. With the increase of the sample size, the stability of both the moment estimation and the maximum likelihood estimation is significantly improved, but the difference between them still exists. Figures 3.1 to 3.3 respectively illustrate the visual results of setting 1, setting 2, and setting 4. It can be observed from the figures that, across all samples, the RMSE values corresponding to MLE are smaller, indicating that MLE has better uracypared to MOM. \\