Advanced Parameter Estimation for the Gompertz-Makeham Process: A Comparative Study of MMLE, PSO, CS, and Bayesian Methods

A research study investigates how to estimate Gompertz-Make ham Process (GMP) parameters within non-homogeneous Poisson processes (NHPP). Authorities have developed Modified Maximum Likelihood Estimation (MMLE) as an improvement over standard Maximum Likelihood Estimation (MLE) to resolve parameter estimation accuracy issues. The study utilizes combination artificial intelligence optimizations through particle swarm optimization (PSO) and cockoo search (CS) alongside Bayesian estimation to assess different methods. This study evaluates MMLE and PSO and CS with Bayesian methods through Root Mean Square Error (RMSE) and Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) statistical accuracy measurements during a simulation analysis. The MMLE estimation technique delivers better estimation precision than PSO, CS and Bayesian methods during the performance assessment. The methodology is validated through its use in modeling operational failures at the Badoush Cement Factory and COVID-19 case occurrences in Italy, showing its capability to model failure rates alongside event occurrences. The research generates progress in NHPP statistical estimation methods which gives a stronger analytical platform for reliability monitoring and survival model prediction and epidemiological projection. Research into the GMP needs to focus on including time-dependent elements and structural dependency mechanisms to enhance the model's capability and guess making power.