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　　\newpage

　　\section{Comparison and Analysis}
　　\vspace{0.5cm}

　　\subsection{Assessment Criteria}

　　As shown in Table II-XV, in this study, we use RMSE as the comparative standard for assessing estimation accuracy. RMSE (Root Mean Square Error) is a statistical measure used to assess the accuracy of parameter estimates in predictive models [21]. It quantifies the average discrepancy between predicted and observed values, crucial for evaluating the goodness-of-fit in regression analysis and time series forecasting [21]. RMSE helps researchers and analysts gauge the precision of parameter estimates, guiding model refinement and selection [3]. It serves as a fundamental tool in various fields, including economics, environmental science, and finance, where precise parameter estimation is essential for decision-making and policy formulation. RMSE ensures robust and reliable parameter estimation, facilitating more accurate predictions and informed decisions [2].

　　The Root Mean Square Error (RMSE) is calculated using the following formula:

　　\begin{equation}
　　RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n}(y_i - \hat{y}_i)^2}
　　\end{equation}

　　where:\\
　　1. n is the number of observations.\\
　　2. $y_i$ is the actual value of the i-th observation.\\
　　3. $\hat{y}_i$ is the estimated value of the i-th observation.\\

　　The formula computes the square root of the average squared differences between the actual and estimated values, providing a measure of the typical deviation of the estimated values from the actual values.

　　\subsection{Comparative 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.\\

　　\noindent1. Sample sizes n: These include 50, 100, 500, and 1000, representing the number of samples in each simulation.\\
　　2. Parameter estimates: Estimates of $\hat{\mu}$, $\hat{\sigma}$, and $\hat{\beta}$ for each sample size according to different true values of $\mu$, $\sigma$, and $\beta$. This study considered seven different assignments for \(\mu\), \(\sigma\), and \(\beta\), which are respectively (3,1,1), (3,1,2), (3,1,3), (3,2,1), (3,3,1), (2,1,1), and (1,1,1). Each assignment is a setting.\\
　　3. RMSE: Root Mean Square Error for each estimate, used to evaluate the accuracy of the estimates.\\
　　4. The number of iterations is \(N = 1000\).\\

　　Here 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 accuracy 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, with most settings converging to the given true values. However, in some cases, there are estimates that deviate significantly from the true values, such as in setting 4 (3,2,1) and setting 5 (3,3,1), where the estimate of \(\hat{\sigma}\) performs poorly.\\

　　\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}
　　In the simulation results, it can be observed that there is some difference between the moment estimation (MOM) and the maximum likelihood estimation (MLE) 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 (n=500, 1000), the stability of both the moment estimation and the maximum likelihood estimation is significantly improved, but the difference between them still exists.\\