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A function that relates a joint cumulative distribution function to the distribution functions of the individual variables. If the individual distribution functions are known, but the joint distribution is unknown, then a copula can be used to suggest a suitable form for the joint distribution.

Let F be the multivariate distribution function for the random variables X1, X2,…, Xn and let the cumulative distribution function of Xj be Fj (for all j). Define random variables U1, U2,…, Un by Uj=Fj(Xj) for each j, so that the marginal distribution of each Uj has a continuous uniform distribution in the interval (0, 1). Assume that for each value uj there is a unique value xj=F−1(uj) and let the joint cumulative distribution function of U1, U2,…, Un be C. Then , for all u1, u2,…, un in (0, 1), since Uj<uj if and only if Xj < Fj-1(uj). The function C is called the copula. An equivalent equation to the above is C{F1(x1), F2(x2),…, Fn(xn)}=F(x1, x2,…, xn), for all x1, x2,…, xn , where uj=Fj(xj) for each j. Sklar's theorem, formulated by Abe Sklar of the Illinois Institute of Technology and published in 1959, states that, for a given F, there is a unique C such that this equation holds.

Note that it may well be that it is not possible to express the inverse functions Fj-1 in a simple form (an example is the multivariate normal distribution).

Assuming that the copula and the marginal distribution functions are differentiable, the corresponding result for probability density functions is that

f(x1, x2,…, xn)=c{F1(x1), F2(x2),…, Fn(xn)} f1(x1)f2(x2)…fn(xn).

The trivial case where c {F1(x1), F(x2),…, F(xn)}=1 corresponds to the case where the nX-variables are independent. Thus the copula encapsulates the interdependencies between the X-variables and is therefore also known as the dependence function. If c(u1, u2,…, un) is the joint probability density function of U1, U2,…, Un, then

c(u1, u2,…, un)=f(x1, x2,…, xn)/{f1(x1)f2(x2)…fn(xn)},

where , for each j.

Subjects: Probability and Statistics.

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