Using the Aquila Optimizer to Estimate the Two Parameters of the Fréchet Distribution

Abstract

The Fréchet distribution is one of the commonly used Extreme Value Distributions (EVDs) in statistical modeling and heavy-tailed data analysis, where it plays an important role in describing product lifetimes as well as climatic and financial phenomena. The estimation of its two parameters, namely the shape parameter and the scale parameter, is traditionally based on the Maximum Likelihood Estimation (MLE) method. However, maximizing the likelihood function for this distribution involves numerical difficulties, which necessitates the use of numerical optimization methods. In this study, we propose the use of the Aquila Optimizer (AO), a recent metaheuristic algorithm inspired by the hunting behavior of eagles, as an efficient numerical tool for maximizing the likelihood function of the Fréchet distribution. The objective function was formulated as the negative log-likelihood function (-LogL), and the Aquila Optimizer was employed to obtain the optimal estimates of the distribution parameters. Several simulation experiments with different sample sizes were conducted to compare the performance of the proposed method with a conventional approach represented by the Nelder–Mead method, using the Mean Squared Error (MSE) criterion. The simulation results demonstrated that the Aquila Optimizer outperformed the Nelder–Mead algorithm in many cases, although the superiority was slight. The results also showed that both algorithms were consistent, as their MSE values decreased with increasing sample size. In addition, a practical application was carried out using real data, and the results of the survival function estimation indicated a good fit.