On the estimation of nested Archimedean copulas: a theoretical and an experimental comparison

A lot of progress regarding estimation of nested Archimedean copulas has been booked since their introduction by Joe (Multivariate models and dependence concepts. Chapman and Hall, London, 1997). The currently published procedures can be seen as particular cases of two different, more general, approaches. In the first approach, the tree structure of the target nested Archimedean copulas is estimated using hierarchical clustering to get a binary tree, and then parts of this binary tree are collapsed according to some strategy. This two-step estimation of the tree structure paves the way for estimation of the generators according to the sufficient nesting condition afterwards, this sufficient nesting condition on the generators ensuring the resulting estimated nested Archimedean copula is a proper copula. In contrast to the first approach, the second approach estimates the tree structure free of any concern for the generators. While this is the main strength of this second approach, it is also its main weakness: estimation of the generators afterwards so that the resulting nested Archimedean copula is a proper copula still lacks a solution. In this paper, both approaches are formally explored, detailed explanations and examples are given, as well as results from a performance study where a new way of comparing tree structure estimators is offered. A nested Archimedean copula is also estimated based on exams results from 482 students, and a naive attempt to check the fit is made using principal component analysis.