第 4 章
\chapter{Introduction}
\begin{adjustwidth}{2.5cm}{1cm}
\small This Chapter explains......
\end{adjustwidth}
\section{Research Background}
Monte Carlo simulations are a powerful tool for evaluating the performance of different parameter estimation methods. By generating numerous synthetic datasets under controlled conditions, researchers can assess the accuracy, bias, and efficiency of each method. The Root Mean Square Error (RMSE) is commonly used as a performance metric in these simulations, providing a clear measure of estimation precision.
\section{Pearson type III distribution}
The Pearson Type III distribution, introduced by Karl Pearson in the early 20th century, is a versatile probability distribution commonly used to model skewed data. Its flexibility in fitting asymmetric data makes it particularly valuable in various fields such as hydrology, finance, environmental science, and engineering. The distribution is characterized by its shape parameter $\beta $ scale parameter $\sigma $ and location parameter $\mu $, which determine its skewness and spread.
\begin{equation}
f(x;\mu,\sigma,\beta) = \frac{1}{\sigma\, \Gamma(\beta)} \left(\frac{x - \mu}{\sigma}\right)^{\beta-1} e^{-\frac{x - \mu}{\sigma}}
\end{equation}\\
where $x$ ranges from $0$ to $ \infty $, and $ \mu $ ranges from $ 0 $ to $ x $.\\
Here is an explanation of each parameter:
\begin{itemize}
\item $x$ represents the random variable.
\item $\mu$ is the location parameter, indicating the horizontal shift of the distribution along the $x$-axis.
\item $\sigma$ is the scale parameter, governing the spread or variability of the distribution.
\item $\beta$ is the shape parameter, determining the shape of the distribution. Higher values of $a$ lead to distributions with heavier tails.
\item $\Gamma(\beta)$ is the gamma function, which is a generalization of the factorial function to real and complex numbers.
\end{itemize}
\vspace{0.5cm}
The corresponding CDF can be expressed as:
\begin{equation}
F(x;\mu,\sigma,\beta) = \int_{\mu}^{x} \frac{1}{\sigma\, \Gamma(\beta)} \left(\frac{t - \mu}{\sigma}\right)^{\beta-1} e^{-\frac{t - \mu}{\sigma}} \, dt
\end{equation}\\
To solve this integral, we often use numerical methods or lookup tables for specific values of $\mu$, $\sigma$, and $\beta$. Alternatively, for some special cases, closed-form expressions for the CDF might be available.
The CDF represents the probability that a random variable $X$ is less than or equal to a certain value $x$.\\
And the mean, variance and skewness are
\begin{equation}
E(X)=\mu + \beta \cdot \sigma
\end{equation}
\begin{equation}
Var(X)=\sigma^2 \cdot \beta
\end{equation}
\begin{equation}
\gamma(X) = \frac{2}{\sqrt{\beta}}
\end{equation}\\
We can plot the pdf of the Pearson type III distribution with $\mu$ = 1, $\sigma$ = 2, and $\beta$ = 3, as shown in Figure 1, with $\mu$ = 2, $\sigma$ = 1, and $\beta$ = 3, as shown in Figure 2 and with $\mu$ = 1, $\sigma$ = 1, and $\beta$ = 3, as shown in Figure 3.\\
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{pdf.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 1, $\sigma$ = 2, and $\beta$ = 3}
\label{LP}
\end{figure}
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{Rplot-2.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 2, $\sigma$ = 1, and $\beta$ = 3}
\label{LP}
\end{figure}
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{Rplot-3.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 1, $\sigma$ = 1, and $\beta$ = 3}
\label{LP}
\end{figure}
In this study, the parameters $\mu$, $\sigma$ and $\beta$ are estimated by applying different parametric estimation methods discussed in the following section.\\
%可以添加以下内容：
%1、Pearson type III distribution与Gamma distribution的关系
%2、简要介绍Pearson家族的其他分布
\begin{adjustwidth}{2.5cm}{1cm}
\small This Chapter explains......
\end{adjustwidth}
\section{Research Background}
Monte Carlo simulations are a powerful tool for evaluating the performance of different parameter estimation methods. By generating numerous synthetic datasets under controlled conditions, researchers can assess the accuracy, bias, and efficiency of each method. The Root Mean Square Error (RMSE) is commonly used as a performance metric in these simulations, providing a clear measure of estimation precision.
\section{Pearson type III distribution}
The Pearson Type III distribution, introduced by Karl Pearson in the early 20th century, is a versatile probability distribution commonly used to model skewed data. Its flexibility in fitting asymmetric data makes it particularly valuable in various fields such as hydrology, finance, environmental science, and engineering. The distribution is characterized by its shape parameter $\beta $ scale parameter $\sigma $ and location parameter $\mu $, which determine its skewness and spread.
\begin{equation}
f(x;\mu,\sigma,\beta) = \frac{1}{\sigma\, \Gamma(\beta)} \left(\frac{x - \mu}{\sigma}\right)^{\beta-1} e^{-\frac{x - \mu}{\sigma}}
\end{equation}\\
where $x$ ranges from $0$ to $ \infty $, and $ \mu $ ranges from $ 0 $ to $ x $.\\
Here is an explanation of each parameter:
\begin{itemize}
\item $x$ represents the random variable.
\item $\mu$ is the location parameter, indicating the horizontal shift of the distribution along the $x$-axis.
\item $\sigma$ is the scale parameter, governing the spread or variability of the distribution.
\item $\beta$ is the shape parameter, determining the shape of the distribution. Higher values of $a$ lead to distributions with heavier tails.
\item $\Gamma(\beta)$ is the gamma function, which is a generalization of the factorial function to real and complex numbers.
\end{itemize}
\vspace{0.5cm}
The corresponding CDF can be expressed as:
\begin{equation}
F(x;\mu,\sigma,\beta) = \int_{\mu}^{x} \frac{1}{\sigma\, \Gamma(\beta)} \left(\frac{t - \mu}{\sigma}\right)^{\beta-1} e^{-\frac{t - \mu}{\sigma}} \, dt
\end{equation}\\
To solve this integral, we often use numerical methods or lookup tables for specific values of $\mu$, $\sigma$, and $\beta$. Alternatively, for some special cases, closed-form expressions for the CDF might be available.
The CDF represents the probability that a random variable $X$ is less than or equal to a certain value $x$.\\
And the mean, variance and skewness are
\begin{equation}
E(X)=\mu + \beta \cdot \sigma
\end{equation}
\begin{equation}
Var(X)=\sigma^2 \cdot \beta
\end{equation}
\begin{equation}
\gamma(X) = \frac{2}{\sqrt{\beta}}
\end{equation}\\
We can plot the pdf of the Pearson type III distribution with $\mu$ = 1, $\sigma$ = 2, and $\beta$ = 3, as shown in Figure 1, with $\mu$ = 2, $\sigma$ = 1, and $\beta$ = 3, as shown in Figure 2 and with $\mu$ = 1, $\sigma$ = 1, and $\beta$ = 3, as shown in Figure 3.\\
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{pdf.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 1, $\sigma$ = 2, and $\beta$ = 3}
\label{LP}
\end{figure}
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{Rplot-2.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 2, $\sigma$ = 1, and $\beta$ = 3}
\label{LP}
\end{figure}
\begin{figure}[ht!] %!t
\centering
\includegraphics[width=3.5in]{Rplot-3.png}
\caption{PDF of Pearson type III distribution with $\mu$ = 1, $\sigma$ = 1, and $\beta$ = 3}
\label{LP}
\end{figure}
In this study, the parameters $\mu$, $\sigma$ and $\beta$ are estimated by applying different parametric estimation methods discussed in the following section.\\
%可以添加以下内容：
%1、Pearson type III distribution与Gamma distribution的关系
%2、简要介绍Pearson家族的其他分布