Discretization of the Inverse Rayleigh-G Family: Theoretical Properties, Machine Learning-Based Parameter Estimation, and Practical Applications

Abstract

This analysis investigates a novel two-parameter discrete distribution, namely the Discrete Inverse Rayleigh Exponential (DIRE) distribution, which is derived from the Inverse Rayleigh-G family using a survival discretization method. The DIRE distribution features adaptable probability mass and hazard rate functions, capable of exhibiting symmetric, asymmetric, monotonic, and reversed-J-shaped behaviors, making it highly suitable for modeling a wide range of real-world data. Key statistical properties, such as mean, variance, moment-generating function, and dispersion index, are thoroughly examined. For parameter estimation, both Maximum Likelihood Estimation (MLE) and a machine learning-based K-Nearest Neighbors (K-NN) algorithm are utilized. Extensive simulations and real-world dataset analyses reveal that the DIRE distribution surpasses existing models in goodness-of-fit metrics, with the K-NN estimator demonstrating superior accuracy and robustness compared to MLE. The practical utility of the DIRE distribution is illustrated through two empirical datasets—COVID-19 case counts and failure time data—highlighting its effectiveness in managing complex discrete data. The results indicate that this new model offers improved flexibility and reliability, making it a valuable tool for statistical modeling and machine learning applications.