Copula

A copula is a function that joins univariate distribution functions to form multivariate distribution functions. Writing , a two-dimensional copula is a function such that

(1)

and

(2)

for all , and

(3)

for all such that and (Nelsen 2006).

More generally, an -dimensional copula is a joint distribution function whose marginal distributions are uniform distributions on . Given univariate distribution functions , ..., , the function

(4)

is a joint distribution function with the prescribed marginal distributions. Conversely, Sklar's theorem states that every joint distribution function has such a representation. The copula is unique when all the marginal distribution functions are continuous functions (Nelsen 2006). This separates the modeling of dependence from the choice of marginal distributions.

For example, choosing

(5)

gives

(6)

The resulting random variables are statistically independent. In general, the statistical distribution with joint distribution function is distinct from that with joint distribution function , whose marginal distributions are uniform distributions.

Statistical distributions constructed from a copula and prescribed marginal distributions, called copula distributions, are implemented in the Wolfram Language as CopulaDistribution[ker, dist1, dist2, ...], where ker specifies the copula kernel and dist1, dist2, ... are the prescribed marginal distributions.