Copula (probability theory)

In probability theory and statistics, a copula is a multivariate probability distribution for which the marginal probability distribution of each variable is uniform. Copulas are used to describe the dependence between random variables. Their name 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.

Consider a random vector ${\displaystyle (X_{1},X_{2},\dots ,X_{d})}$. Suppose its marginals are continuous, i.e. the marginal CDFs ${\displaystyle F_{i}(x)=\mathbb {P} [X_{i}\leq x]}$ are continuous functions. By applying the probability integral transform to each component, the random vector

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