Maximum Likelihood Estimation of the Weighted Mixture Generalized Gamma Distribution
Wikanda Phaphan, Teerawat Simmachan, İbrahim Abdullahi · Mathematics and Statistics · 2023
The three-parameter weighted mixture generalized gamma (WMGG) distribution was developed from the four-parameter mixture generalized gamma (MGG) distribution since the parameter estimation of MGG distribution faced with the problem. The estimate of the weighted parameter p was out of the interval [0, 1]. The previous study proposed the maximum likelihood estimators (MLEs) of the WMGG distribution. However, their MLEs were written in nonlinear equations, and certain iterative methods were necessarily needed to solve numerically. The three parameters λ, β, and α were estimated by the quasi-Newton method. Nevertheless, this method performed well only the parameter λ. This motivated the main objective of this work. Consequently, the parameter estimation of the WMGG was further improved. This article developed two maximum likelihood estimation methods: expectation-maximization (EM) algorithm and simulated annealing algorithm of the three parameters of the WMGG distribution. These two methods were compared to the previous study's quasi-Newton method. Monte Carlo simulation technique was employed to assess the algorithm's performance. Sample sizes ranged from small to large as 10, 30, 50, and 100. The simulation was repeated 10,000 rounds in each scenario. Assessment criteria were the mean square error (MSE) and bias. The results revealed that the EM algorithm outperformed the other methods. Furthermore, the quasi-Newton method had the lowest efficiency.