## 2 Background
### Basic Parameter Estimation Methods for the Generalized Pareto Distribution  
The most critical step in the practical application of the generalized Pareto distribution for modeling the statistical properties of sea clutter is to achieve accurate parameter estimation and to improve the detection performance of sea radar. For different distribution models, a robust parameter estimation method is the basis for the application of the distribution model. The commonly used parameter estimation methods include moment estimation class methods, maximum likelihood estimation methods, quantile parameter estimation methods, and artificial intelligence class parameter estimation methods. Figure 1: Generalized Pareto distribution amplitude PDF curve. Among the moment estimation class methods, both positive 2 nd/4 th-order moment estimation and 1 st-order logarithmic moment estimation have closed expressions for parameter estimation, which can easily obtain the results of parameter estimation, but are limited by the estimation range of shape parameters. The positive 0.5 th/1 st-order moment estimation extends the estimation range of the shape parameter to the fractional domain, but the method does not have a closed expression. The positive 0.25 th-order logarithmic moment estimation method can achieve a further extension of the shape parameter, thus widening the space where the generalized Pareto distribution should be used in the statistical modeling of sea clutter, but the method also cannot derive a closed parameter estimation expression. The maximum likelihood estimation can obtain a relatively high accuracy of parameter estimation, but the parametric results of the scale parameters are given by a nonlinear system of equations, which again does not have a displayed solution. In the following, the mathematical forms of each type of parameter estimation method introduced above will be derived one by one and used as the basis for the optimization of parameter estimation of the generalized Pareto distribution later. The method of moment estimation class parameter estimation equates the moments of each order of the probability distribution model of the sample to the cumulants of the statistical distribution model and obtains all parameter values of the distribution model by solving the system of moment equations of each order. Assuming the unknown parameter \(\theta=(a,b)\) of the generalized Pareto distribution, the \(r\)-th-order moments of the origin of the probability density function can be calculated from Equation (5) as  
\[ \begin{split} E(x^{\prime})&=\int_{0}^{\infty}x^{ \prime}f(x;\theta)\mathrm{d}x,\{r\in R\text{ and }r\
eq 0\}\\ &=\int_{0}^{\infty}\frac{2 abx^{\prime+1}}{\left(1+bx^{2}\right)^{ \left(a+1\right)}}\mathrm{d}x=\frac{a}{b^{2}}\int_{0}^{\infty}\frac{\lambda^ {r/2}}{\left(1+\lambda\right)^{\left(a+1\right)}}\mathrm{d}\lambda\\ &\qquad=\frac{1}{b^{\prime/2}}\frac{\Gamma\left(1+\frac{\xi}{2} \right)\Gamma\left(a-\frac{\xi}{2}\right)}{\Gamma\left(a\right)}\end{split}  \tag{7}\]  
where the variable \(\lambda=bx^{2}\), the solution process involves some of the relevant properties of the gamma function operation, and it is known from its properties that the above equation needs to satisfy \(a>r/2\) to hold, i.e., the estimation range of the shape parameter will be limited. If the sea clutter sample data \(\mathbf{X}=[x_{1},x_{2},\cdots,x_{n}]\) are known, the \(r\)-th-order sample origin moment of the random data \(\mathbf{X}\) is \(m_{r}=1/n\sum_{i}^{n}x_{i}^{r}\). The generalized Pareto distribution contains two parameters to be estimated, so it is necessary to list the equations of two different order moments and complete the equation solution to obtain the determined distribution model. When \(r\) is taken as 2 and 4, respectively, the parameter estimation expression for positive 2 nd/4 th-order moment estimation is:  
\[\left\{ \begin{array}{l}\hat{a}=2+\frac{2m_{2}^{2}}{m_{4}-2m_{2}^{2}}\\ \hat{b}=\frac{1}{\left(\hat{a}-1\right)m_{2}}\end{array} \right.\  
of the parameters.