Nonparametric estimation of the tree structure of a nested Archimedean copula

One of the features inherent in nested Archimedean copulas, also called hierarchical Archimedean copulas, is their rooted tree structure. In this paper, a nonparametric, rank-based method to estimate this structure is developed. Our approach consists in representing the rooted tree structure as a set of trivariate structures that can be estimated individually. Indeed, for any triple of variables there are only four possible rooted tree structures and, based on a sample, a choice can be made by performing comparisons between the three bivariate margins of the empirical distribution of the triple. The set of estimated trivariate structures can then be used to build an estimate of the global rooted tree structure. This approach has the advantage that no assumptions about the nested Archimedean copula is required prior to the estimation of its structure.