Copula Models of Dependence

This chapter introduces the definition of a copula, and discusses the basic properties of copulas. The role of Sklar's theorem is that not just every copula function with marginal distributions as arguments is a valid bivariate distribution. It states that every valid bivariate distribution can be represented as a copula of its marginals. The chapter contains the simplest examples of copula functions. It describes the methods of construction of joint distributions using two most popular subclasses of copula functions: elliptical copulas and Archimedean copulas. One of attractive features of copula models is a natural way to organize simulation from joint distributions, whose dependence structure is defined by copulas. The chapter considers simulation from the elliptical family and Archimedean copulas. Copula models are becoming a popular instrument of applied statistical analysis in such fields as finance, risk management, or engineering.