Construction of a new continuous distribution with two variables and examination of its properties

This thesis introduces a novel class of bivariate generalized Rayleigh distributions derived from latent random variables following the generalized Rayleigh distribution and belonging to the Marshall-Olkin family. Their properties, including marginal and conditional probability density functions, as well as dependence measures such as Kendall's tau and Spearman's rho, are systematically investigated. The parameter estimation process incorporates both MLE and Bayesian methodologies. The Bayesian approach employs advanced Markov Chain Monte Carlo techniques to estimate posterior distributions, while bootstrap methods are utilized to construct robust confidence intervals for these parameters. The framework is validated by applying the proposed models to a real-world dataset, with their performance compared against existing models. Additionally, the thesis extends its contributions to the development of a novel copula that demonstrates improved dependence modeling capabilities. Key measures such as stability and tail behavior are thoroughly evaluated for the new copula. Furthermore, distortion-based extensions are explored, resulting in the creation of new families of univariate and bivariate copulas, which exhibit enhanced flexibility and practical applicability.