A Robust Flexible Hazard Rate Mixture Anchored in a Novel Over-dispersed Count Distribution

This research presents an innovative three-parameter discrete mixed distribution for the modeling of count data in various scientific domains that often display intricate characteristics, including overdispersion, heavy tails, and unimodality. The proposed distribution is created by combining an extended geometric distribution with the discrete Burr-Hatke distribution using a weighted mixing framework. This formulation establishes a unified and adaptable modeling framework wherein the component distributions manifest as distinct limiting cases as the mixing parameter approaches its boundary values. A thorough investigation of the parameter space's identifiability, the distribution's log-convexity, and its asymptotic properties is conducted. Closed forms are used to get various statistical and reliability properties. The hazard rate function is very flexible, allowing for patterns that go up and down, are unimodal, or are shaped like a bathtub. This makes it possible for the model to capture different ways that things can fail in reliability engineering and survival analysis. The maximum likelihood method is used to estimate parameters, and the log-likelihood function and score equations are derived in detail. The performance of the estimators on finite samples is carefully tested through a lot of Monte Carlo simulations with different sample sizes and parameter settings. These simulations show that the estimators are consistent, n-consistent, and asymptotically normal. Evidence-based guidelines for suitable sample sizes are offered to aid practitioners in implementation. Three real-world datasets show that the proposed distribution works well in practice and outperforms other distributions. The proposed model consistently surpasses various established discrete distributions across all applications, as validated by standard information criteria and goodness-of-fit metrics. These results show that it is a strong and useful tool for modeling complex over-dispersed count data that have outliers and extreme values.