Maximum likelihood estimation for the two-parameter Maxwell distribution

The Maxwell distribution is popular in physics, chemistry and statistical dynamics. Since the estimators obtained using the maximum likelihood method have the desired properties of being efficient, consistent, and asymptotically normal under regularity conditions, this method is a widely used method to estimate the parameters of a probability distribution. Although parameter estimates can be obtained using this method, the derivatives of the log-likelihood equations, known as ML estimation equations, with respect to the parameters do not always have clear solutions. Therefore, numerical methods are used to solve these equations. Various traditional numerical methods for this purpose are well-documented in the literature. Additionally, highly powered algorithms with no required mathematical assumption that improve the computational efficiency like heuristic algorithms can be used to solve these equations. In this article, the maximum likelihood method is applied to estimate the location and scale parameters of the two-parameter Maxwell distribution. High-performance heuristic algorithms, such as particle swarm optimization and genetic algorithms, are used and compared with traditional numerical techniques, including Nelder-Mead and Quasi-Newton methods. To show the performance of these techniques, an extensive Monte Carlo simulation study was conducted to compare the efficiencies of maximum likelihood estimators of model parameters concerning bias, mean square error, and deficiency criteria. Simulation results showed that genetic algorithm and particle swarm optimization estimators are more efficient than the other traditional algorithms for estimating the location and scale parameters for the two-parameter Maxwell distribution.