Why copulas have been successful in many practical applications: A theoretical explanation based on computational efficiency - Vladik Kreinovich (2018)
© Springer International Publishing Switzerland 2015. A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) F(x) = Prob(X ≤ x). When several random variables X 1 ,..., X n are independent, the corresponding cdfs F 1 (x 1 ),..., F n (x n ) provide a complete description of their joint distribution. In practice, there is usually some dependence between the variables, so, in addition to the marginals F i (x i ), we also need to provide an additional information about the joint distribution of the given variables. It is possible to represent this joint distribution by a multi-D cdf F(x 1 ,..., x n ) = Prob(X 1 ≤ x 1 & ... & X n ≤ x n ), but this will lead to duplication-since marginals can be reconstructed from the joint cdf-and duplication is a waste of computer space. It is therefore desirable to come up with a duplication-free representation which would still allow us to easily reconstruct F(x 1 ,..., x n ). In this paper, we prove that among all duplication-free representations, the most computationally efficient one is a representation in which marginals are supplements by a copula. This result explains why copulas have been successfully used in many applications of statistics: since the copula representation is, in some reasonable sense, the most computationally efficient way of representing multi-D probability distributions.