Copula-based analysis of asymmetrically distributed joint data under a risk profile using the discrete Type-I extreme value distribution

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.