\text{RMSE}($\hat{\sigma}$) & 0.1564 & 0.1165 & 0.0797 & 0.0707 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of Minimum $\chi^2$ Estimation with Equiprobable Cells for $m=20$}

\centering

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

\hline

$n$ & 50 & 100 & 200 & 400 \\

\hline

$\hat{\mu}$ & 1.0355 & 1.0224 & 1.0151 & 1.0127 \\

$\hat{\sigma}$ & 1.9832 & 1.9793 & 1.9947 & 1.9940 \\

\text{RMSE}($\hat{\mu}$) & 0.0444 & 0.0286 & 0.0193 & 0.0160 \\

\text{RMSE}($\hat{\sigma}$) & 0.1575 & 0.1091 & 0.0714 & 0.0672 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of RP-based Minimum $\chi^2$ Estimation for $m=5$}

\centering

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

\hline

$n$ & 50 & 100 & 200 & 400 \\

\hline

$\hat{\mu}$ & 0.9915 & 0.9922 & 0.9972 & 0.9924 \\

$\hat{\sigma}$ & 2.0220 & 2.0184 & 2.0048 & 2.0062 \\

\text{RMSE}($\hat{\mu}$) & 0.1826 & 0.1242 & 0.0880 & 0.0787 \\

\text{RMSE}($\hat{\sigma}$) & 0.2055 & 0.1434 & 0.1026 & 0.0860 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of RP-based Minimum $\chi^2$ Estimation for $m=10$}

\centering

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

\hline

$n$ & 50 & 100 & 200 & 400 \\

\hline

$\hat{\mu}$ & 0.9871 & 0.9939 & 0.9979 & 0.9965 \\

$\hat{\sigma}$ & 2.0659 & 2.0348 & 2.0217 & 2.0151 \\

\text{RMSE}($\hat{\mu}$) & 0.1033 & 0.0716 & 0.0498 & 0.0451 \\

\text{RMSE}($\hat{\sigma}$) & 0.1842 & 0.1237 & 0.0864 & 0.0790 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of RP-based Minimum $\chi^2$ Estimation for $m=20$}

\centering

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

\hline

$n$ & 50 & 100 & 200 & 400 \\

\hline

$\hat{\mu}$ & 0.9957 & 0.9974 & 0.9980 & 0.9990 \\

$\hat{\sigma}$ & 2.1124 & 2.0639 & 2.0421 & 2.0346 \\

\text{RMSE}($\hat{\mu}$) & 0.0607 & 0.0449 & 0.0314 & 0.0269 \\

\text{RMSE}($\hat{\sigma}$) & 0.2021 & 0.1287 & 0.0906 & 0.0757 \\

\hline

\end{tabular}

\end{table}

\newpage

\subsection{parative Analysis}

As we can see in Tables \ref{table:2.1} and \ref{table:2.2}, when the sample size is small, the uracies of the parametric estimation methods are low, since the RMSEs are nearly 0.3. We are not satisfied with these results, so our samples should be expanded, and the number of our cells should be increased. As the sample size increases, the uracies are getting higher than those when the sample size is small. As \( n = 400 \), the RMSEs are small enough for us to ept.

In addition, we can clearly see from the tables above that moment estimation method and RP-minimum \( \chi^2 \) estimation method are slightly better than the other two methods. In the next chapter, further applications regarding to the four kinds of estimation methods will be implemented. We can observe the differences of the estimation behaviors in a more explicit way as further simulation studies are constructed.