Generalized distributions have been studied a lot recently because of their flexibility and reliability in modeling lifetime data. The two-parameter Exponentially-Modified Logistic distribution is a flexible modified distribution that was introduced in 2018. It is regarded as a strong competitor for widely used classical symmetrical and non-symmetrical distributions such as normal, logistic, lognormal, log-logistic, and others. In this study, the unknown parameters of the Exponentially-Modified Logistic distribution are estimated using the maximum likelihood method. Five meta-heuristic algorithms, including the genetic algorithm, particle swarm optimization algorithm, grey wolf optimization algorithm, whale optimization algorithm, and sine cosine algorithm, are applied in order to solve the nonlinear likelihood equations of the study model. The efficiencies of all maximum likelihood estimates for these algorithms are compared via an extensive Monte Carlo simulation study. The performance of the maximum likelihood estimates for the location and scale parameters of the Exponentially-Modified Logistic distribution developed with the genetic algorithm and grey wolf optimization algorithms is the most efficient among others, according to simulation findings. However, the genetic algorithm is two times faster than grey wolf optimization and can be considered better than grey wolf optimization considering the computation time criterion. Six real datasets are analyzed to show the flexibility of this distribution.

Nakagami dağılımı, radyo sinyallerinin sönümlenmesini modellemek için ortaya çıkmıştır ve çeşitli disiplinlerde yaygın olarak kullanılmaktadır. Bu çalışmada, dağılımın şekil ve ölçek parametrelerini tahmin etmek için en çok olabilirlik (ML) tahmin yöntemi kullanılmıştır. Ancak, bu dağılım için olabilirlik denklemlerinin açık çözümleri bulunmamaktadır. Bu nedenle, bu denklemlerin çözümü için, parçacık sürüsü optimizasyon (PSO), genetik algoritma (GA) ve quasi-newton (QN) algoritmaları olmak üzere üç temel algoritma kullanılmıştır. Bu algoritmaların performanslarının karşılaştırmaları, yan, hata kareler ortalaması (MSE) ve eksiklik (DEF) kriterleri dikkate alınarak, kapsamlı bir Monte-Carlo simülasyon çalışması ile yapılmıştır. Model, kullanışlılığını göstermek amacıyla dört gerçek veri setine uygulanmıştır.

The log-logistic distribution has been widely used in several fields, including engineering, survival analysis, and economics. The method of maximum likelihood estimation is used in this study for estimating the shape and scale parameters for the log-logistic distribution, whereas in the case of the log-logistic distribution, likelihood equations lack explicit solutions. Therefore, problems with solving likelihood equations can be solved by using two highly efficient algorithms, which are the whale optimization algorithm and the Nelder-Mead algorithm, as well as by showing the applicability of this distribution by comparing it with other well-known classical distributions. To demonstrate the performance of each algorithm implemented, an extensive Monte Carlo simulation study has been conducted. The performance of maximum likelihood estimators for each algorithm has been evaluated in terms of mean square error and deficiency criteria. It has been seen that the whale optimization algorithm provides the best estimates for the log-logistic distribution parameters according to the simulation data.

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.

Generalized distributions have been studied a lot recently because of their flexibility and relia-bility in modeling lifetime data. The two-parameter exponentially-modified logistic distribution is a new flexible modified distribution that was introduced recently. It is regarded as a strong competitor for widely used classical symmetrical and non-symmetrical distributions such as normal, logistic, lognormal, and log-logistic. In this study, the unknown parameters of the dis-tribution are estimated using the maximum likelihood method. The Grey Wolf Optimization al-gorithm, which is a new meta-heuristic algorithm, is applied in order to solve the nonlinear like-lihood equations of the study model. The performance of the Grey Wolf Optimization method is compared to that of the other meta-heuristic algorithms used in this study, which include the Whale Optimization Algorithm, the Sine Cosine Algorithm, and the Particle Swarm Optimiza-tion Algorithm. The efficiencies of maximum likelihood estimates for all algorithms are com-pared via an extensive Monte-Carlo simulation study. The likelihood estimates for the location α and scale β parameters of the exponentially-modified logistic distribution developed with the Grey Wolf Optimization algorithm are the most efficient among others, according to simulation findings. Four real datasets are analyzed to show the flexibility of this distribution.