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See also: Modeling correlation introduction, Copulas, Copulas in ModelRisk
An important class of copulas - because of the ease with which they can be constructed and the nice properties they possess - are the Archimedean copulas, which are defined by:
where j is the generator of the copula, which I will explain later. The general relationship between Kendall's tau t and the generator of an Archimedean copula ja(t) for a bivariate data set can be written as:
For example, the relationship between
Kendall's tau 
and the Clayton copula parameter
for a bivariate data set is given by:
The definition doesn't extend to a multivariate data set of n variables because there will be multiple values of tau, one for each pairing. However, one can calculate tau for each pair and use the average, i.e.:
There are three Archimedian copulas in common use: the Clayton, Frank and Gumbel.
The Clayton copula is
an asymmetric Archimedean copula, exhibiting greater dependence in the
negative tail than in the positive. This copula is given by:
And its generator is:
where:
The relationship between Kendall's
tau
and the Clayton copula parameter 
is given by:
This Copula is implemented in ModelRisk as VoseCopulaBiClayton.
The Frank copula is a symmetric Archimedean
copula given by:
And its generator is:
where:
The relationship between Kendall's
tau 
and the Frank copula parameter
is given by:
where
is a Debye function of the first kind.
This Copula is implemented in ModelRisk as VoseCopulaBiFrank.
The Gumbel copula (a.k.a.
Gumbel-Hougard copula) is an asymmetric Archimedean copula, exhibiting
greater dependence in the positive tail than in the negative. This copula
is given by:
And its generator is:
where:
The relationship between Kendall's
tau 
and the Gumbel copula parameter
is given by:
This Copula is implemented in ModelRisk as VoseCopulaBiGumbel.
Read on: Elliptical copulas - Normal and T