Journals Information
Mathematics and Statistics Vol. 13(1), pp. 1 - 11
DOI: 10.13189/ms.2025.130101
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Parameters Estimation of the Gompertz-Makeham Process in Non-Homogeneous Poisson Processes: Using Modified Maximum Likelihood Estimation and Artificial Intelligence Methods
Adel S. Hussain 1, Kawar B. Mahmood 2, Ismat M. Ibrahim 3, Ali F. Jameel 4, Sundas Nawaz 5, Mohammad A. Tashtoush 4,6,*
1 Department of IT, Amedi Technical Institute, Duhok Polytechnic University, Iraq
2 Department of Technical Mechanical Power, College of Engineering, Al-Amarah University, Iraq
3 Department of Dental Technology, Technical College of Duhok, Duhok Polytechnic University, Iraq
4 Faculty of Education and Arts, Sohar University, Sultanate of Oman
5 Department of Mathematical Sciences, Fatima Jinnah Women University, Pakistan
6 Jadara University Research Center, Jadara University, Jordan
ABSTRACT
In this paper, we study the rate of occurrence of the non-homogeneous Poisson process by introducing the Gompertz-Makeham distribution as a rate of occurrence, known as the Gompertz-Makeham Process (GMP). To estimate parameters of this process, we propose the Maximum Likelihood Estimator (MLE) and introduce a modification to address its limitations in finding accurate estimators. The modified method, referred to as the Modified Maximum Likelihood Estimator (MMLE), employs an intelligent algorithm for the likelihood function to improve its performance. We compare the results of MMLE with another intelligent method, Particle Swarm Optimization (PSO), to identify the most effective estimator for the rate of occurrence of the proposed Gompertz-Makeham process. Additionally, this paper includes a simulation study of the process and presents a practical application. By utilizing the MMLE and PSO algorithms, we seek to provide accurate parameter estimation for the Gompertz-Makeham process, thereby enhancing its applicability in diverse domains such as mortality modeling, reliability analysis, and disease progression studies. The comparative analysis between MMLE and PSO offers valuable insights into the performance and effectiveness of intelligent algorithms in estimating the rate of occurrence for NHPP processes. Applied to a real data application, it studies operating periods in days between two successive stops for the raw materials factory from the General Company for Northern Cement / Badoush Cement Factory and estimates the rate for the number of stops for the factory for the period time from 1st April 2020 to 1st January 2022.
KEYWORDS
Artificial Intelligence, Gompertzian-Makeham Process, Modified Maximum Likelihood Estimation, Non-Homogeneous Poisson Processes, Particle Swarm Optimization Algorithm, Simulation
Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Adel S. Hussain , Kawar B. Mahmood , Ismat M. Ibrahim , Ali F. Jameel , Sundas Nawaz , Mohammad A. Tashtoush , "Parameters Estimation of the Gompertz-Makeham Process in Non-Homogeneous Poisson Processes: Using Modified Maximum Likelihood Estimation and Artificial Intelligence Methods," Mathematics and Statistics, Vol. 13, No. 1, pp. 1 - 11, 2025. DOI: 10.13189/ms.2025.130101.
(b). APA Format:
Adel S. Hussain , Kawar B. Mahmood , Ismat M. Ibrahim , Ali F. Jameel , Sundas Nawaz , Mohammad A. Tashtoush (2025). Parameters Estimation of the Gompertz-Makeham Process in Non-Homogeneous Poisson Processes: Using Modified Maximum Likelihood Estimation and Artificial Intelligence Methods. Mathematics and Statistics, 13(1), 1 - 11. DOI: 10.13189/ms.2025.130101.