## 1 Introduction
In response to the problem of non-closed parameter estimation equations in some parameter estimation methods for the generalized Pareto distribution model, this study proposes the use of the particle swarm optimization (PSO) algorithm to search for non-closed expressions and solve the parameter estimation problem. The PSO algorithm is applied to nonlinear optimization problems, including the non-closed expressions of parameter estimation methods such as 0.5 th/1 st-order moment estimation, 0.25 th-order logarithmic moment estimation, and maximum likelihood estimation. Furthermore, to address the limitations of low estimation accuracy, restricted estimation range, and non-closed expressions in traditional parameter estimation methods for the generalized Pareto distribution, this study aims to find an optimal parameter estimation model for the generalized Pareto distribution that does not rely on complex mathematical expressions. To achieve this, a hybrid particle swarm optimization algorithm (HPSO) is proposed to perform the optimal parameter search task for the target objectives in parameter estimation of the generalized Pareto distribution. The main work and innovations of this study are summarized as follows:  
* In response to the non-closed expression phenomenon in different parameter estimation methods, this study investigates the construction of fitness functions for PSO algorithm and HPSO when solving target optimization problems. * By using simulated random data samples of the generalized Pareto distribution, the impact of parameters such as population size and iteration count in the PSO algorithm on its performance is examined, and the optimal parameter configuration for each targeted objective is determined. * A goodness-of-fit test experiment is conducted on two sets of high-intensity ocean wave clutter measured data to compare the fitting performance of the generalized Pareto distribution models obtained through different parameter estimation methods. * The performance of the HPSO algorithm and the PSO algorithm is compared using simulated data based on the parameter estimation fitness function constructed in this study. The experimental results demonstrate that the PSO algorithm not only expands the parameter estimation range of the generalized Pareto distribution but also significantly improves its fitting performance to observed data, particularly in the case of heavy-tailed ocean clutter data. Furthermore, simulation experiments reveal that the HPSO algorithm outperforms the PSO algorithm when it comes to solving the task of optimal parameter search for the generalized Pareto distribution. The remaining sections of the paper are organized as follows. Section 2 provides background information on the generalized Pareto distribution model and its fundamental parameter estimation techniques. In Section 3, we present a comprehensive explanation of the parameter estimation methods for the generalized Pareto distribution model, with a specific focus on the PSO-based approach and the proposed HPSO method. Section 4 presents the experimental results, followed by analysis and discussion. Finally, in Section 5, we present the conclusions drawn from our findings and discuss potential avenues for future research.