\begin{enumerate}

\item Step 1: Determine the true values for $v (v > 2)$, $\mu$, and $\sigma$.

\item Step 2: Generate $m$ cells with respect to the standard Student’s $t$ distribution, denoted by $J_i$, where $i = 1, \ldots, m$.

\item Step 3: Randomly take $n$ samples following the generalized Student’s $t$-distribution with parameters $v (v > 2)$, $\mu$, and $\sigma$.

\item Step 4: Calculate $\hat{\mu}$ by applying the formula in (2.17). Use Newton-Raphson method to solve equation (2.18) numerically for $\hat{\sigma}$ (The ”fsolve” built-in function in MATLAB could be applied).

\item Step 5: Transform all the ”$J_i$”s linearly by $\Delta_i = \hat{\mu} + \hat{\sigma} J_i$, which means that to transform the endpoints of ”$J_i$”s linearly except for infinite endpoints.

\item Step 6: Determine all of the elements specified in (2.23), and solve the equation numerically by applying the ”fsolve” built-in function in MATLAB.

\item Step 7: Repeat steps 3 to 6 for $N$ times, and calculate the RMSE by applying the formula in (2.24).

\end{enumerate}

In step 2, we have two ways of generating cells from the standard Student’s $t$-distribution with degree of freedom $v (v > 2)$. The first way is the Fisher’s classification, i.e., we let $p_1(\theta) = \ldots = p_m(\theta) = \frac{1}{m}$, where $p_i(\theta)$ is defined as what (2.20) has illustrated. The endpoints can be solved when the above equations are satisfied. Hence, the cell intervals are constructed. The second way is the RP classification. From [6], we can obtain the RPs of the standard Student’s $t$-distribution with degree

of freedom $v (v > 2)$, which are denoted as $R_1, \ldots, R_m$. The cell intervals,

$\Delta_1 = (-\infty, \frac{R_1 + R_2}{2})$, $\Delta_j = (\frac{R_{j-1} + R_j}{2}, \frac{R_j + R_{j+1}}{2})$, where $j = 2, \ldots, m - 1$, and

$\Delta_m = (\frac{R_{m-1} + R_m}{2}, \infty)$, are hence constructed.

\subsection{Estimation Results of Simulation}

The estimation results are listed in the following tables. In our simulation study, we set $n = 50, 100, 200, 400$, $m = 5, 10, 20$, $\beta = 1$, $\mu = 1$, $\sigma = 2$, and $N = 1000$. It is worth noticing that the true values of the parameters are selected without loss of generality. The aim of our simulation study is topare the estimation methods and choose the best ones. Hence, the true values of the parameters should be manipulated to be the same.\\

\begin{table}[!htbp]

\caption{Simulation Results of Moment Estimation}

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 0.9080 & 0.9732 & 0.9904 & 0.9932 \\

$\hat{\sigma}$ & 1.9603 & 1.9893 & 1.9963 & 1.9970 \\

RMSE($\hat{\mu}$) & 0.2823 & 0.1561 & 0.0909 & 0.0755 \\

RMSE($\hat{\sigma}$) & 0.4373 & 0.2329 & 0.1302 & 0.1065 \\

\hline

\end{tabular}

\end{table}

\begin{table}[!htbp]

\caption{Simulation Results of MLE}

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 1.1744 & 1.0946 & 1.0463 & 1.0201 \\

$\hat{\sigma}$ & 2.2571 & 2.1283 & 2.0585 & 2.0215 \\

RMSE($\hat{\mu}$) & 0.2324 & 0.1399 & 0.0845 & 0.0529 \\

RMSE($\hat{\sigma}$) & 0.3754 & 0.2350 & 0.1528 & 0.0931 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of Minimum $\chi^2$ Estimation with Equiprobable Cells for $m=5$}

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 1.0684 & 1.0493 & 1.0330 & 1.0299 \\

$\hat{\sigma}$ & 1.9415 & 1.9539 & 1.9628 & 1.9678 \\

\text{RMSE}($\hat{\mu}$) & 0.0850 & 0.0641 & 0.0427 & 0.0375 \\

\text{RMSE}($\hat{\sigma}$) & 0.1782 & 0.1281 & 0.0948 & 0.0817 \\

\hline

\end{tabular}

\end{table}

\begin{table}[htbp]

\caption{Simulation Results of Minimum $\chi^2$ Estimation with Equiprobable Cells for $m=10$}

\centering

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

\hline

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

\hline

$\hat{\mu}$ & 1.0488 & 1.0303 & 1.0222 & 1.0189 \\

$\hat{\sigma}$ & 1.9623 & 1.9732 & 1.9799 & 1.9834 \\

\text{RMSE}($\hat{\mu}$) & 0.0598 & 0.0380 & 0.0278 & 0.0244 \\