## 4 Results and Discussion
### Simulation Experiment Analysis
The particle population search space determined by the fitness function 3 is theoretically within the whole positive real number domain, but the experimental test found that the particle movement to the position where the scale parameter is less than 0.15 results in a false-dead state, which causes the particle population to be trapped in this position all the time during the iterative process. After analysis, it is found that the possible reason for the above phenomenon is that the likelihood function, after taking logarithmic derivatives, limits the range of the scale parameter. To address this problem, we consider expanding the one-dimensional search space of the maximum likelihood function to two dimensions, i.e., directly searching the extremes of the maximum likelihood function in the space of scale parameters and shape parameters, thus skipping the step of taking logarithmic derivatives. Figure 8 gives the iterative process of searching for the maximum likelihood function in different dimensions for the particle population. In which the one-dimensional space of the likelihood function, the search range of the particles is expanded from the original [0.15, +\(\infty\)] to (0, +\(\infty\)]. As a comparison, in the two-dimensional search space of the likelihood function, the search range of the particles is expanded from the original [0.  
hood function, the search range of the particle population is also set to \((0,+\infty]\). As can be seen from the figure, after expanding the search range of particles in the one-dimensional space, the particle population is initially trapped in an undefined position, making the particles inactive and falsely dead. The above phenomenon does not occur when the particle population is searched in the two-dimensional space of the likelihood function, and a higher accuracy of parameter estimation is obtained by searching directly in the two-dimensional space. Although the particle search in two dimensions can solve the problem of one-dimensional search, the expansion of the spatial search domain requires a larger number of particles and iterations to complete the search task, which undoubtedly increases the computational time complexity. Table 2 lists the time consumed by each parameter method of the generalized Pareto distribution. From the data in Table 2, it can be seen that the parameter estimation of positive 1 st-order logarithmic moments is the most efficient, and the parameter estimation time of positive 0.5 th/1 st-order moment estimation, positive 0.25 th-order moment estimation, and maximum likelihood estimation based on the swarm subgroup optimization to solve the non-closed expression is comparable to that of positive 2 nd/4 th-order moment estimation; so the introduction of the particle swarm search does not significantly increase the computational time complexity of the parameter estimation method itself. The two-dimensional space search of the maximum likelihood estimation method doubles its computation time compared to the one-dimensional space, which is also consistent with the previous analysis. The operational efficiency of the algorithm is particularly important in applications with real-time processing, and the various advantages and disadvantages can be weighed in choosing a specific parameter estimation method based on the above discussion, which leads to the selection of a suitable parameter estimation method. The estimation ranges of shape parameters for each parameter estimation method are detailed in Section 2. Here, to investigate the effect of shape parameters on the parameter estimation performance, we fixed the sample length of the simulation data at \(10^{4}\), and the scale parameter \(b\) was set to 0.3. The estimation performance of the two-parameter model was then calculated by varying the shape parameter \(a\) from 0.1 to 10 in equal intervals. The results were evaluated using the relative root mean square error (RRMSE). Figure 9 shows