A flexible bivariate skewed odd-G family of distributions for various Hazard rate profiles in dispersed trends: Theory-driven interdisciplinary joint data analysis

The individual features, temporality, and analysis goals of the random variables determine the distinct challenges that statistical modeling with combined datasets frequently entails. When it comes to areas like stress testing and preventative maintenance, where knowing how reliable a system is and when it will fail is vital, these problems become much more complex. More flexible statistical tools are required in these situations because traditional methods may not work. To deal with the complexities of binary data, especially in cases of imbalance, overdispersion, and changing risk patterns, this study presented two novel bivariate probability models. The Marshall-Olkin shock framework provides the basis for one model, while the Farlie-Gumbel-Morgenstern technique is the basis for the other. Both are mathematically sound and have practical uses since they provide exact analytical formulas for important functions including quantiles, hazard rates, survival functions, and joint probabilities. We focused on two versions that show how these two model families evolved theoretically and how they might be applied in specific cases. Systems where success or failure is represented by binary outcomes across time are practical examples, as are instances where stress levels directly effect the probability of failure. Utilizing comprehensive simulation experiments as support, we employed maximum likelihood estimation to derive the parameters of the models. We then tested the models on three real-world binary datasets, demonstrating their robustness and adaptability in situations where more conventional approaches may fail to do justice to the complexities of the data.