Count data modeling and its practical applications have garnered significant attention in recent research, owing to its relevance in a wide range of fields. This study specifically explores a novel discrete distribution characterized by two parameters, which is derived using the survival discretization method. The statistical properties of this distribution are thoroughly explained in closed forms, with several key mathematical attributes also derived. These characteristics underscore the distribution's effectiveness in modeling data that exhibit (right-skewed) asymmetry and have extended heavy tails, making it particularly suitable for such real-world applications. Furthermore, the failure rate function corresponding to this distribution is particularly appropriate for scenarios characterized by an increasing or bathtub-shaped failure rate over time. The model is also highly versatile, offering valuable insights into probabilistic modeling for datasets that display over dispersion, under dispersion, or equi dispersion. The study introduces several estimation techniques, including the maximum product of spacings, Anderson–Darling, right–tail Anderson–Darling, maximum likelihood estimation, least squares, weighted least squares, Cramer–Von–Mises, and percentile methods. Each of these methods is explained in detail, providing a comprehensive understanding of their application. A ranking simulation study is conducted to evaluate the performance of these estimators across varying sample sizes, using ranking techniques to identify the most effective estimator in different scenarios. The analysis of real-world datasets from biotechnology and industrial engineering further demonstrates the practical utility and relevance of the proposed model. The results highlight the model's ability to offer accurate and insightful analyses, reinforcing its significance in count data modeling and its wide-ranging applications.

Copulas provide a flexible framework for building bivariate probability models that reflect specific dependency patterns. This work introduces a discrete form of the Type-I extreme value (Gumbel) distribution within a copula-based structure. Key mathematical and statistical characteristics are examined, including the joint probability mass function, survival function, hazard rate, conditional expectation, joint probability generating function, and dependency properties such as positive quadrant dependence and total positivity of order two. The bivariate discrete Gumbel model demonstrates strong performance in handling asymmetric data and proves particularly useful for capturing extreme and outlier observations. Its joint hazard rate function adds further flexibility, making it suitable for modeling a range of failure rate behaviors. Parameter estimation is carried out using the maximum likelihood method, and a thorough simulation study evaluates the bias and mean squared errors across various sample sizes. To illustrate its practical relevance, the model is applied to three different real-world datasets: football match outcomes, nasal drainage severity scores, and lens defects involving surface and interior faults.