Why copulas have been successful in many practical applications: A theoretical explanation based on computational efficiency

• Main Authors: Vladik Kreinovich, Hung T. Nguyen, Songsak Sriboonchitta, Olga Kosheleva
• Format: Conference Proceeding
• Published: 2018
• Subjects: Agricultural and Biological Sciences
• Online Access: https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84951019499&origin=inward; http://cmuir.cmu.ac.th/jspui/handle/6653943832/44812
• Institution: Chiang Mai University

© 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.