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

Résumé

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.