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31、031

　　In statistics, bias represents the average difference between an estimate and the true value, and efficiency represents the degree of variation in the estimate [2]. Moment estimates and maximum likelihood estimates may differ in terms of bias and efficiency. Maximum likelihood estimates typically have smaller biases, especially in large sample sizes, but it can be computationally more expensive [3]. Moment estimates can have large biases, especially for small sample sizes, but the calculation is relatively simple [3].\\

　　The conclusions above were confirmed in the results of this study. Firstly, among the 7 different settings, Maximum Likelihood Estimation (MLE) demonstrated better stability compared to the Method of Moments (MOM). For example, Figure 4 presents a visual analysis of the results for setting 1, where the solid line represents the MLE results. It can be clearly seen from the figure that the RMSE values corresponding to MLE are consistently smaller across all sample sizes. Indeed, our objective is to minimize the RMSE of the estimates as much as possible because smaller RMSE indicates less discrepancy between the estimated and true values, thus indicating the effectiveness of the model.

　　Tables 3 and 4 respectively display the simulation results corresponding to MOM for settings 2 and 3. It is evident from these tables that when \(n=50\), the difference between \(\hat{\beta}\) and the true values (\(\beta=2\) and \(3\)) is significant, resulting in very large RMSE values. However, the corresponding MLE estimates (Tables 0 and 11) are much more stable, indicating that the method of moments is not suitable in this scenario. Therefore, selecting the appropriate parameter estimation method based on the actual scenario is crucial for obtaining more accurate estimates.

　　Based on the aforementioned discussion and analysis, in situations with small sample sizes such as \(n=50, 100\), we should preferably use Maximum Likelihood Estimation for parameter estimation. However, in cases of large sample sizes, such as \(n \geq 1000\), due to economic and time costs, we can choose between the method of moments and maximum likelihood estimation.

　　Furthermore, according to the results, both MOM and MLE performed poorly in estimating \(\sigma\) in settings 4 and 5, indicating the presence of volatility in parameter estimation. Particularly, when sample sizes are small (e.g., 50 or 100), estimates of \(\hat{\sigma}\) and \(\hat{\beta}\) exhibit significant fluctuations.\\

　　To further analyze these data, the following additional analyses can be conducted:\\

　　\noindent1. Analysis of variance: To examine the influence of sample size on the accuracy of parameter estimates.\\
　　2. Graphical representation: Plotting estimates of \(\hat{\mu}\), \(\hat{\sigma}\), and \(\hat{\beta}\) along with their RMSE values for different \(n\) values to visualize the impact of sample size on estimate stability.\\
　　3. Regression analysis: Analyzing the specific effect of sample size on RMSE and the trend of estimates as sample size varies.

　　\begin{figure}[ht!] %!t
　　\centering
　　\includegraphics[width=3.5in]{setting2.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 $\beta$ = 2}
　　\label{LP}
　　\end{figure}

　　\begin{figure}[ht!] %!t
　　\centering
　　\includegraphics[width=3.5in]{setting4.png}
　　\caption{The variation trend of each parameter estimators and RMSE with sample size of MOM and MLE in setting1: $\mu$ = 3, $\sigma$ = 2, and $\beta$ = 1}
　　\label{LP}
　　\end{figure}
　　\vspace{0.5cm}