  A large number of commonly used parametric Archimedean copula (AC) families
are restricted to a single parameter, connected to a concordance measure such
as Kendall's tau. This often leads to poor statistical fits, particularly in
the joint tails, and can sometimes even limit the ability to model concordance
or tail dependence mathematically. This work suggests outer power (OP)
transformations of Archimedean generators to overcome these limitations. The
copulas generated by OP-transformed generators can, for example, allow one to
capture both a given concordance measure and a tail dependence coefficient
simultaneously. For exchangeable OP-transformed ACs, a formula for computing
tail dependence coefficients is obtained, as well as two feasible OP AC
estimators are proposed and their properties studied by simulation. For
hierarchical extensions of OP-transformed ACs, a new construction principle,
efficient sampling and parameter estimation are addressed. By simulation,
convergence rate and standard errors of the proposed estimator are studied.
Excellent tail fitting capabilities of OP-transformed hierarchical AC models
are demonstrated in a risk management application. The results show that the OP
transformation is able to improve the statistical fit of exchangeable ACs,
particularly of those that cannot capture upper tail dependence or strong
concordance, as well as the statistical fit of hierarchical ACs, especially in
terms of tail dependence and higher dimensions. Given how comparably simple it
is to include OP transformations into existing exchangeable and hierarchical AC
models, this transformation provides an attractive trade-off between
computational effort and statistical improvement.
