  Explicit functional forms for the generator derivatives of well-known
one-parameter Archimedean copulas are derived. These derivatives are essential
for likelihood inference as they appear in the copula density, conditional
distribution functions, or the Kendall distribution function. They are also
required for several asymmetric extensions of Archimedean copulas such as
Khoudraji-transformed Archimedean copulas. Access to the generator derivatives
makes maximum-likelihood estimation for Archimedean copulas feasible in terms
of both precision and run time, even in large dimensions. It is shown by
simulation that the root mean squared error is decreasing in the dimension.
This decrease is of the same order as the decrease in sample size. Furthermore,
confidence intervals for the parameter vector are derived. Moreover, extensions
to multi-parameter Archimedean families are given. All presented methods are
implemented in the open-source R package nacopula and can thus easily be
accessed and studied.
