HPCOPULA Procedure
Archimedean Copulas
Overview of Archimedean Copulas
Let function 
be a strict Archimedean copula generator function, and suppose that 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 HPCOPULA procedure are the Clayton copula, the Frank copula, and the Gumbel copula.
Simulation
Suppose that the generator of the Archimedean copula is 
. Then the simulation method that uses a Laplace-Stieltjes transformation of the distribution function is given by Marshall and Olkin (1988), where 
:
Generate a random variable V that has the distribution function F such that
. Draw samples from the independent uniform random variables
. Return
.
The Laplace-Stieltjes transformations are as follows:
For the Clayton copula,
, and the distribution function F is associated with a gamma random variable that has a shape parameter of 
and a scale parameter of 1. For the Gumbel copula,
, and F is the distribution function of the stable variable 
, where 
. For the Frank copula where
, 
, and 
is a discrete probability function 
. This probability function is related to a logarithmic random variable that has a parameter value of 
.
For more information about simulating a random variable from a stable distribution, see Theorem 1.19 in Nolan (2010). For more information about simulating a random variable from a logarithmic series, see Chapter 10.5 in Devroye (1986).
For a Frank copula where 
and 
, the simulation can be done through conditional distributions as follows:
Draw independent
from a uniform distribution. Let
. Let
.