Enhanced Parameter Estimation for the Modified Gompertz-Makeham Model in Nonhomogeneous Poisson Processes Using Modified Likelihood and Swarm Intelligence Approaches

Abstract

This research introduces a new method to estimate Nonhomogeneous Poisson Processes (NHPP) parameters through integration of Modified Gompertz-Makeham Process (MGMP) with statistical methods and optimization techniques. The Modified Maximum Likelihood Estimator (MMLE) constitutes our proposed estimation solution because traditional Maximum Likelihood Estimation (MLE) proves inadequate when analyzing complex systems and precise parameter assessment particularly under conditions of low sample counts or nonlinear conditions. The method adds structural constraints together with supplementary information to the likelihood function to achieve enhanced inference accuracy. The highly non-linear likelihood equations require solution through the implementation of Particle Swarm Optimization (PSO) as a biologically based metaheuristic algorithm which shows excellence when dealing with challenging parameter spaces. The combination of MMLE with PSO results in a hybrid framework which demonstrates its performance against standard techniques through simulated data series and factory failure records taken from the Badush Cement Factory. The MMLE-PSO method proves its superior accuracy levels proven by tests utilizing RMSE, MSE, PRR and PP metrics. The study presents compelling evidence for the combination of statistical models with intelligent optimization methods which improves reliability studies and survival analytics and failure prediction. The research findings present statistical validity alongside computational feasibility which provides an effective parameter estimation methodology for changing stochastic processes.