Copulas

In today’s data-driven world, the analysis of multiple variables and their joint behavior has become increasingly important. Copulas offer a versatile and powerful framework for understanding and modeling the dependence structure between variables. This chapter serves as a guide to the concept of copula, highlighting its main properties and key aspects discussed in each section. In Sect. 1.1, we delve into the motivation behind studying copulas and their broad range of applications in various fields. We recognize the significant contribution of Abe Sklar to copula theory, emphasizing his pivotal role in introducing copulas to the research community. Section 1.2 then focuses on the fundamental concepts of copula theory, starting with their definition as bivariate distribution functions with uniform margins. The properties of copulas, such as the rectangular inequality and the boundary conditions, are explored. Sklar’s theorem, which establishes the relationship between a bivariate distribution and its copula, is introduced as a key result. In Sect. 1.3, we discuss the simplest copulas and their properties. We explore the concept of elliptical copulas, which are a special class of copulas based on elliptical distributions. Additionally, we delve into different measures of dependence used to quantify the strength of the relationship between variables. Section 1.6 expands the discussion from bivariate to multivariate copulas. We explore the extension of copulas to handle dependence among multiple variables, allowing for a more comprehensive analysis of complex probabilistic models.