Parametric Quantile Copula Regression for Bivariate Median Based Unit Rayleigh

This research introduces a comprehensive parametric copula regression framework, meticulously designed to model the joint behavior of two response variables, specifically conditioned on a chosen predictor variable. The innovative methodology effectively synthesizes parametric regression techniques for the marginal distributions with a copula function, which adeptly captures the intricate dependence structure between the response variables. Leveraging Sklar’s theorem, the formulation of the joint conditional density is achieved by expressing it as the product of the marginal densities and a copula density. Notably, the parameter of the copula is allowed to vary as a function of the covariate, enhancing the model's flexibility and adaptability. A Gumbel copula is utilized, which is particularly adept at modeling upper-tail dependence, and its parameter is intricately linked to the predictor through a smooth parametric function, ensuring a nuanced representation of relationships. For parameter estimation, the study employs the Inference Functions for Margins (IFM) method, which facilitates efficient and accurate estimation of model parameters. Additionally, a range of diagnostic tools based on the analysis of residuals are utilized to validate the adequacy of the model, ensuring that the fit is not only statistically sound but also meaningful in practical applications. The findings of this study reveal a significant trend: the dependence between the response variables tends to diminish as the predictor variable increases. This observation highlights the critical need to address associations that depend on covariates, a factor often overlooked in traditional analyses. Overall, this proposed framework stands out as a versatile and interpretable tool for analyzing complex multivariate relationships. Its potential for application spans various fields, making it a valuable resource in both academic research and professional practice. The framework offers possibilities for deeper insights into data that involves multiple interconnected response variables, reinforcing its relevance across diverse analytical contexts.