Tails of multivariate Archimedean copulas

TL;DR

This paper provides a comprehensive, user-friendly classification of the tail behaviors of Archimedean copulas, enabling better model selection and construction based on tail dependence properties.

Contribution

It introduces a decision tree for classifying copula tails, computes tail quantities for many families, and offers methods to construct models with specific tail behaviors.

Findings

Classifies tails into asymptotic dependence and independence categories.

Provides tail quantities for numerous single-parameter families.

Enables construction of models with tailored tail dependence structures.

Abstract

A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.