Densities of nested Archimedean copulas

TL;DR

This paper derives a general formula for the density of nested Archimedean copulas, enabling likelihood-based inference and practical evaluation in high-dimensional settings.

Contribution

It provides the first general, tractable formula for the density of nested Archimedean copulas, including efficient computation methods.

Findings

Derived a general formula for derivatives of nested Archimedean copulas.

Presented a tractable density formula for arbitrary dimensions.

Developed an efficient method for evaluating the log-density.

Abstract

Nested Archimedean copulas recently gained interest since they generalize the well-known class of Archimedean copulas to allow for partial asymmetry. Sampling algorithms and strategies have been well investigated for nested Archimedean copulas. However, for likelihood based inference it is important to have the density. The present work fills this gap. A general formula for the derivatives of the nodes and inner generators appearing in nested Archimedean copulas is developed. This leads to a tractable formula for the density of nested Archimedean copulas in arbitrary dimensions if the number of nesting levels is not too large. Various examples including famous Archimedean families and transformations of such are given. Furthermore, a numerically efficient way to evaluate the log-density is presented.