COPULA Procedure

Sklar’s Theorem

The copula models are tools for studying the dependence structure of multivariate distributions. The usual joint distribution function contains the information both about the marginal behavior of the individual random variables and about the dependence structure between the variables. The copula is introduced to decouple the marginal properties of the random variables and the dependence structures. An m-dimensional copula is a joint distribution function on left-bracket 0 comma 1 right-bracket Superscript m with all marginal distributions being standard uniform. The common notation for a copula is upper C left-parenthesis u 1 comma ellipsis comma u Subscript m Baseline right-parenthesis.

The Sklar (1959) theorem shows the importance of copulas in modeling multivariate distributions. The first part claims that a copula can be derived from any joint distribution functions, and the second part asserts the opposite: that is, any copula can be combined with any set of marginal distributions to result in a multivariate distribution function.

  • Let F be a joint distribution function and upper F Subscript j Baseline comma j equals 1 comma ellipsis comma m, be the marginal distributions. Then there exists a copula upper C colon left-bracket 0 comma 1 right-bracket Superscript m Baseline right-arrow left-bracket 0 comma 1 right-bracket such that

    upper F left-parenthesis x 1 comma ellipsis comma x Subscript m Baseline right-parenthesis equals upper C left-parenthesis upper F 1 left-parenthesis x 1 right-parenthesis comma ellipsis comma upper F Subscript m Baseline left-parenthesis x Subscript m Baseline right-parenthesis right-parenthesis

    for all x 1 comma ellipsis comma x Subscript m Baseline in left-bracket negative normal infinity comma normal infinity right-bracket. Moreover, if the margins are continuous, then C is unique; otherwise C is uniquely determined on Ran upper F 1 times midline-horizontal-ellipsis times Ran upper F Subscript m Baseline, where Ran upper F Subscript j Baseline equals upper F Subscript j Baseline left-parenthesis left-bracket negative normal infinity comma normal infinity right-bracket right-parenthesis is the range of upper F Subscript j.

  • The converse is also true. That is, if C is a copula and upper F 1 comma ellipsis comma upper F Subscript m Baseline are univariate distribution functions, then the multivariate function defined in the preceding equation is a joint distribution function with marginal distributions upper F Subscript j Baseline comma j equals 1 comma ellipsis comma m.

Last updated: June 19, 2025