Why Copulas Have Been Successful in Many Practical Applications: A Theoretical Explanation Based on Computational Efficiency

A natural way to represent a 1-D probability distribution is to store its cumulative distribution function (cdf) \(F(x)=\mathrm{Prob}(X\le x)\). When several random variables \(X_1,\ldots ,X_n\) 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 \( F(x_1,\ldots ,x_n)=\mathrm{Prob}(X_1\le x_1\, \& \,\ldots \, \& \,X_n\le x_n)\), 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 \(F(x_1,\ldots ,x_n)\). 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.

Authors and Affiliations

1. Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

   Vladik Kreinovich

2. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

   Hung T. Nguyen

3. Faculty of Economics, Chiang Mai University, Chiang Mai, Thailand

   Hung T. Nguyen & Songsak Sriboonchitta

4. University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

   Olga Kosheleva

Corresponding author

Correspondence to Vladik Kreinovich .

Editors and Affiliations

1. Science and Technology, Japan Advanced Institute of, Nomi, Japan

   Van-Nam Huynh

2. Science, Dept of Systems Innovation, Osaka University,Graduate School of, Osaka, Japan

   Masahiro Inuiguchi

3. Université de Technologie de Compiègne, Compiègne, France

   Thierry Demoeux

Reprints and permissions

© 2015 Springer International Publishing Switzerland

Policies and ethics