Let function be a strict Archimedean copula generator function and suppose its inverse is completely monotonic on . A strict generator is a decreasing function that satisfies and . A decreasing function is completely monotonic if it satisfies
An Archimedean copula is defined as follows:
The Archimedean copulas available in the COPULA procedure are the Clayton copula, the Frank copula, and the Gumbel copula.
Suppose the generator of the Archimedean copula is . Then the simulation method using LaplaceStieltjes transformation of the distribution function is given by Marshall and Olkin (1988) where :
Generate a random variable with the distribution function such that .
Draw samples from independent uniform random variables .
Return .
The LaplaceStieltjes transformations are as follows:
For the Clayton copula, , and the distribution function is associated with a Gamma random variable with shape parameter and scale parameter one.
For the Gumbel copula, , and is the distribution function of the stable variable with .
For the Frank copula with , , and is a discrete probability function . This probability function is related to a logarithmic random variable with parameter value .
For details about simulating a random variable from a stable distribution, see Theorem 1.19 in Nolan (2010). For details about simulating a random variable from a logarithmic series, see Chapter 10.5 in Devroye (1986).
For a Frank copula with and , the simulation can be done through conditional distributions as follows:
One method to estimate the parameters is to calibrate with Kendall’s tau. The relation between the parameter and Kendall’s tau is summarized in the following table for the three Archimedean copulas.
Table 10.2: Calibration Using Kendall’s Tau
Copula Type 

Formula for 

Clayton 


Gumbel 


Frank 

No closed form 
In Table 10.2, for , and for . In addition, for the Frank copula, the formula for has no closed form. The numerical algorithm for root finding can be used to invert the function to obtain as a function of .
Alternatively, you can use the MLE or the CMLE method to estimate the parameter given the data and . The loglikelihood function for each type of Archimedean copula is provided in the following sections.
Fitting the Clayton Copula
For the Clayton copula, the loglikelihood function is as follows (Cherubini, Luciano, and Vecchiato, 2004, Chapter 7):
Let be the derivative of . Then the first order derivative is
The second order derivative is
Fitting the Gumbel Copula
A different parameterization is used for the following part, which is related to the fitting of the Gumbel copula. For Gumbel copula, you need to compute . It turns out that for ,
where is a function that is described later. The copula density is given by
where , , ,,, and .
The log density is
Now the first order derivative of the log density has the decomposition
Some of the terms are given by
where
The last term in the derivative of the is
Now the only remaining term is , which is related to . Wu, Valdez, and Sherris (2007) show that satisfies a recursive equation
with .
The preceding equation implies that is a polynomial of x and therefore can be represented as
In addition, its coefficient, denoted by , is a polynomial of . For simplicity, use the notation . Therefore,
Fitting the Frank copula
For the Frank copula,
When , a Frank copula has a probability density function
where .
The log likelihood is
Denote
Then the derivative of the log likelihood is
The term in the last summation is
The function satisfies a recursive relation
with . Note that is a polynomial whose coefficients do not depend on ; therefore,
where
For the case of and , the bivariate density is