Copula (probability theory)

In probability theory and statistics, a copula is a multivariate cumulative distribution function for which the marginal probability distribution of each variable is uniform on the interval [0, 1]. Copulas are used to describe/model the dependence (inter-correlation) between random variables. Their name, introduced by applied mathematician Abe Sklar in 1959, comes from the Latin for "link" or "tie", similar but unrelated to grammatical copulas in linguistics. Copulas have been used widely in quantitative finance to model and minimize tail risk and portfolio-optimization applications. Sklar's theorem states that any multivariate joint distribution can be written in terms of univariate marginal distribution functions and a copula which describes the dependence structure between the variables. Copulas are popular in high-dimensional statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginals and copulae separately. There are many parametric copula families available, which usually have parameters that control the strength of dependence. Some popular parametric copula models are outlined below. Two-dimensional copulas are known in some other areas of mathematics under the name permutons and doubly-stochastic measures. Consider a random vector . Suppose its marginals are continuous, i.e. the marginal CDFs are continuous functions. By applying the probability integral transform to each component, the random vector has marginals that are uniformly distributed on the interval [0, 1]. The copula of is defined as the joint cumulative distribution function of : The copula C contains all information on the dependence structure between the components of whereas the marginal cumulative distribution functions contain all information on the marginal distributions of . The reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions.