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

Hofert, Marius; Mächler, Martin; McNeil, Alexander J.

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 25-63.

Les pratiques du moment en gestion quantitative du risque nous amènent à étudier des modèles de dépendance en grandes dimensions construits à partir de copules Archimédiennes. La performance d’un grand nombre d’estimateurs, dont plusieurs sont originaux, est étudiée et des solutions sont proposées pour les problèmes numériques associés. Des estimateurs fondés sur le méthode des moments et le beta de Blomqvist ainsi que le tau de Kendall sont comparés à des estimateurs du minimum de distance, du maximum de vraisemblance, du maximum de vraisemblance simulé et du maximum de vraisemblance fondé sur la diagonale de la copule. Des simulations sont réalisées dans le cas où les marges sont connues et dans le cas où elles sont estimées non paramétriquement, en faibles et en grandes dimensions, pour plusieurs familles de copules Archimédiennes. Toutes les méthodes étudiées sont implantées dans le package R copula distribué sous une licence libre. Les solutions numériques présentées dans ce travail restent valident dans le cas de généralisations asymétriques des copules Archimédiennes et d’importantes quantités associées comme la fonction de distribution de Kendall.

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-moments-like 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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