\end{equation}

\begin{equation}

\frac{\partial l(\theta)}{\partial \beta}=-n \log \sigma-n \psi(\beta)+\sum_{i=1}^n \log \left(x_i-\mu\right)=0

\end{equation}

where, $\psi(\beta)=\frac{\partial \ln \Gamma(\beta)}{\partial \beta}$.\\

ording to (3.12), (3.13) and (3.14), we cannot get the algebraic solutions for the above three parameters. In this case, we can construct a simulation study to calculate the estimator $\hat{\mu}$, $\hat{\sigma}$ and $\hat{\beta}$. The details of the simulation procedures are discussed in the following content.

\subsection{Procedures of Estimations}

The Monte Carlo simulation process of estimation for the three parameters in the Pearson type III distribution can be divided into the following steps:

\noindent\textbf{Step 1} Determine the true values for $\mu$, $\sigma$ and $\beta$.\\

\textbf{Step 2} Randomly take n samples following the Pearson type III distribution with parameters $\mu$, $\sigma$ and $\beta$.\\

\textbf{Step 3} Calculate $\hat{\mu}$, $\hat{\sigma}$ and $\hat{\beta}$ by applying the formulas in (3.7), (3.8) and (3.9) for moment estimation, however, for MLE, we need an algorithm to solve equations (3.12), (3.13) and (3.14).\\

\textbf{Step 4} Repeat steps 2 and 3 for N times, all of the above processes are performed in R 4.3.1.\\

The following table includes the detailed specification for each step:

\begin{table}[h]

\centering

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

\hline

Sample sizes \(n\) & 50, 100, 500, and 1000 \\

\hline

Seven different assignments for \(\mu\), \(\sigma\), and \(\beta\) & (3,1,1), (3,1,2), (3,1,3), \\

& (3,2,1), (3,3,1), (2,1,1), and (1,1,1) \\

\hline

Iterations & \(N = 1000\) \\

\hline

\end{tabular}

\end{table}

\subsection{Estimation Results of Simulation}

Tables $3.1-3.7$ present the results of moment estimation, while Tables $3.8-3.14$ present the results of Maximum Likelihood Estimation. In next subsection, we conduct a detailedparative analysis of these results.\\

\begin{table}[!htbp]

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

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 2.6429 & 2.7982 & 2.9539 & 2.976 \\

$\hat{\sigma}$ & 0.8136 & 0.8876 & 0.9759 & 0.9885 \\

$\hat{\beta}$ & 2.3482 & 1.6643 & 1.1393 & 1.0722 \\

RMSE($\hat{\mu}$) & 0.524 & 0.3342 & 0.1411 & 0.1025 \\

RMSE($\hat{\sigma}$) & 0.4584 & 0.3783 & 0.2059 & 0.1498 \\

RMSE($\hat{\beta}$) & 2.5026 & 1.193 & 0.3803 & 0.256 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

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

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 2.1958 & 2.5977 & 2.907 & 2.9524 \\

$\hat{\sigma}$ & 0.862 & 0.917 & 0.9809 & 0.9918 \\

$\hat{\beta}$ & 6.0924 & 3.3176 & 2.2813 & 2.1342 \\

RMSE($\hat{\mu}$) & 1.6038 & 0.764 & 0.3152 & 0.2157 \\

RMSE($\hat{\sigma}$) & 0.4727 & 0.3795 & 0.2099 & 0.1445 \\

RMSE($\hat{\beta}$) & 20.3582 & 2.7083 & 0.8195 & 0.5199 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

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

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 1.1751 & 2.3175 & 2.853 & 2.9107 \\

$\hat{\sigma}$ & 0.8698 & 0.9073 & 0.9781 & 0.9855 \\

$\hat{\beta}$ & 49.9423 & 5.3878 & 3.4273 & 3.2425 \\

RMSE($\hat{\mu}$) & 9.5964 & 1.3937 & 0.4825 & 0.3379 \\

RMSE($\hat{\sigma}$) & 0.4831 & 0.3695 & 0.1977 & 0.1433 \\

RMSE($\hat{\beta}$) & 889.324 & 5.7321 & 1.2264 & 0.8147 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

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

\centering

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

\hline

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

\hline