Copulas are distribution functions with standard uniform univariate marginals. Copulas are widely used for studying dependence among continuously distributed random variables, with applications in finance and quantitative risk management; see, e.g., the pricing of collateralized debt obligations (Hofert and Scherer, Quantitative Finance, 11(5), 775-787, 2011). The ability to model complex dependence structures among variables has recently become increasingly popular in the realm of statistics, one example being data mining (e.g., cluster analysis, evolutionary algorithms or classification). The present work considers an estimator for both the structure and the parameters of hierarchical Archimedean copulas. Such copulas have recently become popular alternatives to the widely used Gaussian copulas. The proposed estimator is based on a pairwise inversion of Kendall's tau estimator recently considered in the literature but can be based on other estimators as well, such as likelihood-based. A simple algorithm implementing the proposed estimator is provided. Its performance is investigated in several experiments including a comparison to other available estimators. The results show that the proposed estimator can be a suitable J Intell Inf Syst alternative in the terms of goodness-of-fit and computational efficiency. Additionally, an application of the estimator to copula-based Bayesian classification is presented. A set of new Archimedean and hierarchical Archimedean copula-based Bayesian classifiers is compared with other commonly known classifiers in terms of accuracy on several well-known datasets. The results show that the hierarchical Archimedean copula-based Bayesian classifiers are, despite their limited applicability for high-dimensional data due to expensive time consumption, similar to highly-accurate classifiers like support vector machines or ensemble methods on low-dimensional data in terms of accuracy while keeping the produced models rather comprehensible.

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.

Several successful approaches to structure determination of hierarchical Archimedean copulas (HACs) proposed in the literature rely on agglomerative clustering and Kendall's correlation coe cient. However, there has not been presented any theoretical proof justifying such approaches. This work lls this gap and introduces a theorem showing that, given the matrix of the pairwise Kendall correlation coe cients corresponding to a HAC, its structure can be recovered by an agglomerative clustering technique.

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