What Are Copulas?

Abstract

This chapter introduces a concept for describing the dependence structure between random variables with arbitrary marginal distribution functions. The main idea is to describe the probability distribution of a d-dimensional random vector by two separate objects: (i) the set of univariate probability distributions for all d components, the so-called ‘marginals’, and (ii) a ‘copula’, which is a d-variate function that contains the information about the dependence structure between the components. Although such a separation into marginals and a copula (if done carelessly) bears some potential for irritations (see Section 7.2 and [Mikosch (2006)]), it can be quite convenient in many applications. The rest of this chapter is organized as follows. Section 1.1 presents two examples which motivate the necessity for the use of a copula concept. Section 1.2 presents Sklar’s Theorem, which can be seen as the ‘fundamental theorem of copula theory’.