Archimedean Copulas in High Dimensions: Estimators and Numerical Challenges Motivated by Financial Applications

Marius Hofert, Martin Mächler, Alexander J. McNeil


The study of Archimedean dependence models in high dimensions is motivated by current practice in
quantitative risk management. The performance of known and new parametric estimators for the parameters of
Archimedean copulas is investigated and related numerical difficulties are addressed. In particular, method-of-momentslike
estimators based on pairwise Kendall’s tau, a multivariate extension of Blomqvist’s beta, minimum distance
estimators, the maximum-likelihood estimator, a simulated maximum-likelihood estimator, and a maximum-likelihood
estimator based on the copula diagonal are studied. Their performance is compared in a large-scale simulation study
both under known and unknown margins (pseudo-observations), in small and high dimensions, under small and large
dependencies, and various different Archimedean families. High dimensions up to one hundred are considered and
computational problems arising from such large dimensions are addressed in detail. All methods are implemented in
the open source R package copula and can thus be easily accessed and studied. The numerical solutions developed
in this work extend to various asymmetric generalizations of Archimedean copulas and important quantities such as
distributions of radial parts or the Kendall distribution function.

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Ce travail est autorisé sous licence avec la Licence de paternité Creative Commons 3.0.

SFdS / SMF - Journal de la Société Française de Statistique - ISSN 2102-6238