The Basic Idea
Consider a random vector . Suppose its margins are continuous, i.e. the marginal CDFs are continuous functions. By applying the probability integral transform to each component, the random vector
has uniform margins.
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.
The importance of the above is that the reverse of these steps can be used to generate pseudo-random samples from general classes of multivariate probability distributions. That is, given a procedure to generate a sample from the copula distribution, the required sample can be constructed as
The inverses are unproblematic as the were assumed to be continuous. The above formula for the copula function can be rewritten to correspond to this as:
Read more about this topic: Copula (probability Theory)
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