## 2 Background
### Basic Parameter Estimation Methods for the Generalized Pareto Distribution
From Equation (9), it can be seen that there is no closed expression for the estimation of the shape parameters, so the parameter estimation results cannot be calculated directly by substituting the values of the sample moments. In addition, the method of estimating log moments of the generalized Pareto distribution proposed in the literature [32] also belongs to the moment estimation class of parameter estimation methods, assuming \(Z=x^{2}\), then the \(r\)-th-order origin moments of the intensity probability density function of the generalized Pareto distribution and the corresponding sample moments can be defined as  
\[\left\{ \begin{array}{l}E(z^{r})=\int_{0}^{\infty}z^{r}\ln z\cdot p(z;\theta) \mathrm{d}z\\ \\ \kappa_{r}=\frac{1}{n}\sum\limits_{i=1}^{n}z_{i}^{r}\ln z_{i}\end{array} ,r\in R\right. \tag{10}\]  
Replacing the overall moment with the sample moment in Equation (10) yields  
\[\left\{ \begin{array}{l}\frac{\kappa_{r}}{m_{2r}}-\kappa_{0}=\psi(1+r)-\psi(1) +\psi(\hat{a})-\psi(\hat{a}-r)\\ \\ \hat{b}=\exp(\psi(1)-\kappa_{0}-\psi(a))\end{array} \right. \tag{11}\]  
where \(\Psi(\cdot)\) is the digamma function, which is obtained by taking the logarithm of the gamma function and then deriving it, and satisfies \(\Psi(1+x)=\Psi(x)+1/x\). By the nature of the digamma function, it is known that the effective estimation range of the shape parameter is \(a>r\). When \(r\) is equal to 1, the parameter estimation expression for positive 1 st-order logarithmic moment estimation is  
\[\left\{ \begin{array}{l}\hat{a}=1+\frac{1}{\kappa_{1}/m_{2}-\kappa_{0}-1}\\ \hat{b}=\exp(\psi(1)-\kappa_{0}-\psi(\hat{a}))\end{array} \right. \tag{12}\]  
The positive 1 st-order logarithmic moment estimation exists for the displayed solution, and the parameter estimation can be obtained by calculating the logarithmic moment and the integer order moment values of the corresponding order of the sample and then substituting them into Equation (12). When \(r\) is equal to 0.25, the logarithmic moment estimation corresponds to the positive 0.25 th-order logarithmic moment estimation method, and it can be seen from Equation (11) that the shape parameter estimator does not have a closed expression at this time, and the parameters cannot be solved directly through the substitution of sample moments. Maximum likelihood estimation is a relatively common parameter estimation method with specific applications in the estimation of parameters of various types of statistical distribution models. The method has a relatively high accuracy of parameter estimation and is close to the lower bound of what can be achieved in terms of parameter estimation accuracy. The log-likelihood function of the generalized Pareto distribution is as follows:  
\[ \begin{array}{l}L(a,b)=\ln\prod\limits_{i=1}^{n}f_{x}(x_{i};a,b)=\ln\left[(2 ab)^{n}\prod\limits_{i=1}^{n}\frac{x_{i}}{\left(1+bx_{i}^{2}\right)^{a+1}}\right] \\ \\ =n\ln(2 ab)-\sum\limits_{i=1}^{n}\left(a+1\right)\ln\left(1+bx_{i}^{2}\right) +\sum\limits_{i=1}^{n}\ln x_{i}\end{array}  \tag{13}\]  
To find the optimal parameters when the log-likelihood function is maximized, Equation (13) is derived for the generalized Pareto distribution for the shape parameter \(a\) and the scale parameter \(b\), respectively, and the derivative is set to zero, resulting in the following equation of the likelihood function:\[\left\{ \begin{array}{l}\frac{\partial L(a,b)}{\partial a}=\frac{n}{a}- \sum\limits_{i=1}^{n}\ln(1+bx_{i}^{2})=0\\ \frac{\partial L(a,b)}{\partial b}=\frac{n}{b}-(a+1)\sum\limits_{i=1}^{n}\frac {x_{i}^{2}}{bx_{i}^{2}+1}=0\end{array} \right. \tag{14}\]  
By eliminating the scale parameter \(b\) in the above set of equations through covariance substitution, the maximum likelihood function estimator of the shape parameter \(a\) can be obtained, and it is re-substituted into Equation (14) to obtain the estimator of the scale parameter. The maximum likelihood estimator of the generalized Pareto distribution is as follows:  
\[\left\{