Penalized Estimation of Shannon and Past Entropy in Geometric Distribution Based on Upper Record Values

The geometric distribution, celebrated for its memory-less property and constant failure rate, is a cornerstone in reliability analysis, life testing, and the modelling of discrete-time phenomena. This paper explores the penalized estimation of Shannon and Past Entropy using complete samples and upper record values derived from the geometric distribution. This study effectively captures extreme observations in discrete time settings using upper record values. This approach significantly improves the reliability and accuracy of entropy estimation. Leveraging the Maximum Penalized Likelihood Estimation (MPLE) technique, the research introduces regularization to balance model complexity and data fidelity, mitigating overfitting even in scenarios with limited or highly variable data. The proposed method can be beneficial in applying reliability analysis, identification of anomalies, risk evaluation, and predictive work. This work enriches the broader field of statistical modelling and offers valuable insights for diverse scientific disciplines.