Two-Dimensional Probability Models for the Weighted Discretized Fréchet–Weibull Random Variable with Min–Max Operators: Mathematical Theory and Statistical Goodness-of-Fit Analysis

This study introduces two bivariate extensions of the recently proposed weighted discretized Fréchet–Weibull distribution, termed as bivariate weighted discretized Fréchet–Weibull (BWDFW) distributions. These models are specifically designed for analyzing two-dimensional discrete datasets and are developed using two distinct structural approaches: the minimum operator (BWDFW-I) and the maximum operator (BWDFW-II). A rigorous mathematical formulation is presented, encompassing the joint cumulative distribution function, joint probability mass function, and joint (reversed) hazard rate function. The dependence structure of the models is investigated, demonstrating their capability to capture positive quadrant dependence. Additionally, key statistical measures, including covariance, Pearson’s correlation coefficient, Spearman’s rho, and Kendall’s tau, are derived using the joint probability-generating function. For robust statistical inferences, the parameters of the proposed models are estimated via the maximum likelihood estimation method, with extensive simulation studies conducted to assess the efficiency and accuracy of the estimators. The practical applicability of the BWDFW distributions is demonstrated through their implementation in two real-world datasets: one from the aviation sector and the other from the security and safety domain. Comparative analyses against four existing discrete bivariate Weibull extensions reveal the superior performance of the BWDFW models, with BWDFW-I (minimum operator based) exhibiting greater flexibility and predictive accuracy than BWDFW-II (maximum operator based). These findings underscore the potential of the BWDFW models as effective tools for modeling and analyzing bivariate discrete data in diverse applied contexts.