An approach to structure determination and estimation of hierarchical Archimedean Copulas and its application to Bayesian classification

Abstract

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 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.