A weighted hybrid discrete probability model: Mathematical framework, statistical analysis, estimation techniques, simulation-based ranking, and goodness-of-fit evaluation for over-dispersed data

Data modeling played a crucial role in a variety of research domains due to its widespread practical applications, especially when handling complex datasets. This study explored a specific discrete distribution, characterized by a single parameter, developed using the weighted combining discretization method. The statistical properties of this distribution were rigorously derived and expressed mathematically, covering essential aspects such as moments, skewness, kurtosis, covariance, index of dispersion, order statistics, entropies, mean residual life, residual coefficient of variation function, stress-strength models, and premium principles. These properties highlighted the model's suitability for analyzing right-skewed data with heavy tails, making it a powerful tool for probabilistic modeling in situations where data exhibited overdispersion and increasing failure rates. The research introduced a range of estimation techniques, including maximum product of spacings, method of moments, Anderson-Darling, right-tail Anderson-Darling, maximum likelihood, least squares, weighted least squares, Cramer-Von-Mises, and percentile, each explained in detail. A ranking simulation study was performed to assess the performance of these estimators, with ranking techniques used to determine the most effective estimator across various sample sizes. The study further applied the proposed model to real-world datasets, demonstrating its ability to address complex data scenarios and showcasing its superior performance in comparison to traditional models such as the geometric, Poisson, and negative binomial distributions. Overall, the results emphasized the proposed model's potential as a versatile and effective tool for modeling over-dispersed and skewed data, with promising implications for future research in diverse fields.