## 4 Results and Discussion  
### Simulation Experiment Analysis  
The Monte Carlo method is used to generate random sequences obeying the generalized Pareto distribution. The parameters of the distribution model were set to \(a=2.5\), \(b=0.3\), and the length of the simulated data was \(5\times 10^{4}\). Figure 3 shows the statistical characteristics of the simulated data, from which it can be seen that the statistical characteristics of the simulated data match the theoretical distribution. To study the properties of the three fitness functions using the simulation data, various sample moments of the simulation data are calculated first, and then the results are substituted into Equation (19), that is, the curves of the fitness functions can be obtained as shown in Figure 4.  
The fitness functions 1 and 2 correspond to the non-closed expressions for positive 0.5 th/1 st-order moment estimation and positive 0.25 th-order logarithmic moment estimation, respectively. The fitness function 3 is significantly different from the first two functions because the horizontal coordinate of this function corresponds to the scale parameter domain and corresponds to the non-closed expression of the maximum likelihood estimate of the scale parameter. As can be seen from Figure 4, the range of values of the fitness functions 1 and 2 are both restricted to a certain space, [0.5, +\(\infty\)] and [0.25, +\(\infty\)], respectively, which must match the range of values of the shape parameters of the two-parameter estimation methods. The notches of the extreme value points of these three functions are relatively deep, so the particle population can quickly find the optimal location point  
Figure 4: The curve of the fitness function. Figure 3: Statistical histogram of simulation data. All three particle populations search for the extremes of the fitness function in a one-dimensional space. The fitness functions 1 and 2 have similar curve shapes, so the same algorithm parameter settings can be used. As can be seen in Figures 5 and 6, the deep notches in the curve of the fitness function allow the particles to move quickly to the vicinity of the extreme value point and thus determine the location of the optimal parameters. The increase in the number of particles and the number of iterations will lead to a longer algorithm operation time, but too few particles or insufficient iterations will easily make the particle population unable to complete the established search task, so the reasonable setting of the algorithm parameters is the key to successfully obtaining the optimal parameter estimation results of the distribution model. Too many particles reduce the efficiency of the algorithm while not bringing additional benefits to the population due to the increase in inert particles. From Figure 7, it can be seen that with the increase in the number of iterations, the optimal particle of fitness function 3 appears to sink at the extreme value point. After analysis, it is found that there are infinitely many location points near the extreme value point, and the lowest valley shown in the figure is not the minimum of the curve, so the optimal particle will keep sinking. When the particle is near the notch, the difference in its corresponding horizontal coordinate position is already very small, so there is no need to make the particle sink into the deepest part of the notch after a large number of iterations. In the final experimental test, the number of particles and iterations of the three populations are set to 5 and 10, respectively, which meets the requirements of the particle population search task. The final algorithm parameter settings of this experiment can complete the search task for the optimal position of each fitness function in the shortest time.