## 4 Results and Discussion
### Performance Analysis of Parameter Estimation with HPSO Algorithm
(**a**,**b**) denote the estimation errors of the corresponding parameters when the shape and scale parameters are varied at equal intervals, respectively. Figure 13: Comparison of two different algorithms’ iterative processes. was employed to perform the optimal parameter search task for the fitness function of parameter estimation in the generalized Pareto distribution. To compare the performance differences of various parameter estimation methods for the generalized Pareto distribution after introducing the HPSO algorithm, this experiment employed the 2 nd/4 th-moment estimation (2/4 th-MoM), 1 st-order logarithmic moment estimation (1 st-order ZlogZ), and maximum likelihood estimation (MLE) methods. The known parameters were estimated for simulated random sequences of the generalized Pareto distribution, and the results of these three commonly used parameter estimation methods were compared with the HPSO algorithm. The scale parameter b was fixed at 0.5, and the number of clutter samples was \(5\times 10^{4}\). The selection parameter for each set of clutter data varied evenly from 0.1 to 10, with 10 independent repeated experiments conducted. The curves of mean squared deviation (MSD) fitting results for each parameter estimation method were obtained, as shown in Figure 15a. Similarly, by selecting the scale parameter \(a\) as 3.5 and varying the selection parameter for each set of clutter data from 0.1 to 10, the variation of the MSD fitting results for each parameter estimation method under different scale parameters was studied (Figure 15b). Observing Figure 15a, it can be observed that the differences in MSD fitting curves among the parameter estimation methods gradually decrease as the shape parameter increases. In the range where the shape parameter a is less than 2, the HPSO algorithm demonstrates significantly better MSD fitting performance than the 2 nd/4 th-MoM estimation and the 1 st-order ZlogZ estimation method, approaching the fitting performance of the MLE method. Figure 15b shows that as the scale parameter increases, the MSD fitting curves for all four parameter estimation methods gradually decrease. In addition, the MSD test of the MLE method exhibits significant fluctuations in the range of scale parameters from 1 to 2. Moreover, in the region where the scale parameter b is greater than 2, the MSD test values are consistently more than \(10^{-2}\) higher than other estimation methods. This indicates that the MLE method's fitting performance is poorer than the HPSO algorithm, the 2 nd/4 th-MoM estimation methods, and the 1 st-order ZlogZ estimation method. Based on the results presented in Figure 15, it can be concluded that compared to the other three commonly used parameter estimation methods, the PDF curve of the generalized Pareto distribution approximated by the HPSO algorithm exhibits better robustness and does not suffer from restricted parameter estimation range issues. This algorithm provides a new solution approach for accurately estimating the parameters of the generalized Pareto distribution. The computational time complexity of the parameter estimation methods is also an indicator for evaluating their performance. Table 6 lists the running times of each  
Figure 15: Parameter estimation error curves of two different algorithms. (**a**,**b**) denote the fitting errors when the shape and scale parameters are varied at equal intervals, respectively. parameter estimation method under the same conditions. From Table 6, it can be observed that the 2 nd/4 th-MoM estimation and the 1 st-order ZlogZ estimation method achieve fast parameter estimation processes. The MLE method requires the longest estimation time. Both the PSO algorithm and the HPSO algorithm have the same computational time complexity. Therefore, considering overall performance, the proposed HPSO algorithm demonstrates promising effectiveness in solving the parameter estimation problem for the generalized Pareto distribution.