On structure, family and parameter estimation of hierarchical Archimedean copulas

Abstract

Research on structure determination and parameter estimation of hierarchical Archimedean copulas (HACs) has so far mostly focused on the case in which all appearing Archimedean copulas belong to the same Archimedean family. The present work addresses this issue and proposes a new approach for estimating HACs that involve different Archimedean families. It is based on employing goodness-of-fit test statistics directly into HAC estimation. The approach is summarized in a simple algorithm, its theoretical justification is given and its applicability is illustrated by several experiments, which include estimation of HACs involving up to five different Archimedean families.

Kendall’s tau and agglomerative clustering for structure determination of hierarchical Archimedean copulas

“…Assuming λ( ) and λ( ) to belong to a one-parametric family of generators (or possibly to two di erent one-parametric families of generators), which is a common assumption in many applications of HACs, for a lot of families, it follows that τ = τ implies that C λ( ) = C λ( ) , e.g., cf. Tables 1, 2 and 3 in [3]. Hence, applying this assumption, it follows that…”

“…There, the approach denoted kt_kagg, originally proposed in [2] and merged with an approach to collapsing of HAC structures from [14], has shown the best results in the ability to estimate the true structure of a HAC among the 11 estimators considered. Also note that in [3], the kt_kagg approach was improved by a new approach to collapsing HAC structures and re-estimation of the parameters, which is con rmed in the simulation study reported therein. In the same article, the authors also show that a structure obtained by this improved kt_kagg satis es the su cient nesting condition, which guarantees that a proper copula results, under relatively weak conditions provided the families of the Archimedean generators in the structure are all the same.…”

“…However, for a HAC with all generators from the same one-parametric Archimedean family, e.g., Ali-Mikhail-Haq, Clayton, Frank, Gumbel, Joe or many others, (10) combined with relatively weak assumptions turns to a su cient nesting condition, as has been shown in Lemma 4.1 in [3]. In other words, under the assumption of the same family (e.g., one of the families mentioned above) for all generators in a HAC, one can aim to satisfy (10) instead of (6) in order to construct a proper copula, which is successfully applied in the construction of HAC estimators proposed in [3].…”

“…The formal de nition of HACs we will recall in this work is taken from [3], which is mainly motivated by the fact that it explicitly refers to the tree HAC structure through concepts from graph theory and thus no auxiliary tools to formalize HAC structures are needed. Note that in the following, #s denotes the number of elements of a set s.…”

“…E.g., in [14], the pair of random variables corresponding to the highest value of Kendall's tau is clustered and the new cluster is represented by its Kendall transformation. In [1], the authors suggest to use the diagonal transformation instead of the Kendall's one. In [10], Kendall's tau used in the previous two approaches is replaced by a more general τ-Euclidean metric.…”