### Journals Information

**
Mathematics and Statistics Vol. 11(3), pp. 516 - 527 DOI: 10.13189/ms.2023.110307 Reprint (PDF) (316Kb) **

## Maximum Likelihood Estimation of the Weighted Mixture Generalized Gamma Distribution

**Wikanda Phaphan ^{1}^{,2}, Teerawat Simmachan ^{3}^{,4}^{,*}, Ibrahim Abdullahi ^{5}**

^{1}Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand

^{2}Research Group in Statistical Learning and Inference, KMUTNB, Bangkok 10800, Thailand

^{3}Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani, Thailand

^{4}Thammasat University Research Unit in Data Learning, Thammasat University, Pathum Thani, Thailand

^{5}Department of Mathematics and Statistics, Faculty of Science, Yobe State University, Damaturu, Nigeria

**ABSTRACT**

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.

**KEYWORDS**

Mixture Generalized Gamma Distribution, Expectation-Maximization Algorithm, Simulated Annealing, Quasi-Newton Method

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Wikanda Phaphan , Teerawat Simmachan , Ibrahim Abdullahi , "Maximum Likelihood Estimation of the Weighted Mixture Generalized Gamma Distribution," Mathematics and Statistics, Vol. 11, No. 3, pp. 516 - 527, 2023. DOI: 10.13189/ms.2023.110307.

(b). APA Format:

Wikanda Phaphan , Teerawat Simmachan , Ibrahim Abdullahi (2023). Maximum Likelihood Estimation of the Weighted Mixture Generalized Gamma Distribution. Mathematics and Statistics, 11(3), 516 - 527. DOI: 10.13189/ms.2023.110307.