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

) = 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.