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

Górecki, Jan; Hofert, Marius; Holeňa, Martin

doi:10.1515/demo-2017-0005

Cited by 13 publications

(7 citation statements)

References 19 publications

(39 reference statements)

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“…As the structure estimator in Górecki et al (2016Górecki et al ( , 2017c does not, by contrast to Okhrin et al (2013a), require any assumptions on the family of the nested ACs, it is thus immediately feasible also for HOPAC estimation. Moreover, there exist theoretical justifications for such an estimator -given a HAC, Okhrin, Okhrin and Schmid (2013b) show that its structure can be uniquely recovered from all its bivariate margins, and Theorem 2 in Górecki, Hofert and Holeňa (2017b) show that it is possible just from all its pairwise Kendall's coefficients. Finally, as this estimator, formalized by Algorithm 1 in Górecki et al (2017c) (see also our Appendix), showed the best results in the ratio of successfully estimated true HAC structures on the basis of simulation studies, see Górecki et al (2016) or Uyttendaele (2017), we adopt it to our HOPAC estimation approach.…”

Section: Estimating Hopacsmentioning

confidence: 99%

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

Górecki¹,

Hofert²,

Okhrin³

2020

Preprint

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A large number of commonly used parametric Archimedean copula (AC) families are restricted to a single parameter, connected to a concordance measure such as Kendall's tau. This often leads to poor statistical fits, particularly in the joint tails, and can sometimes even limit the ability to model concordance or tail dependence mathematically. This work suggests outer power (OP) transformations of Archimedean generators to overcome these limitations. The copulas generated by OP-transformed generators can, for example, allow one to capture both a given concordance measure and a tail dependence coefficient simultaneously. For exchangeable OP-transformed ACs, a formula for computing tail dependence coefficients is obtained, as well as two feasible OP AC estimators are proposed and their properties studied by simulation. For hierarchical extensions of OP-transformed ACs, a new construction principle, efficient sampling and parameter estimation are addressed. By simulation, convergence rate and standard errors of the proposed estimator are studied. Excellent tail fitting capabilities of OP-transformed hierarchical AC models are demonstrated in a risk management application. The results show that the OP transformation is able to improve the statistical fit of exchangeable ACs, particularly of those that cannot capture upper tail dependence or strong concordance, as well as the statistical fit of hierarchical ACs, especially in terms of tail dependence and higher dimensions. Given how comparably simple it is to include OP transformations into existing exchangeable and hierarchical AC models, this transformation provides an attractive trade-off between computational effort and statistical improvement.

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Section: Estimating Hopacsmentioning

confidence: 99%

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

Górecki¹,

Hofert²,

Okhrin³

2020

Preprint

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“…This algorithm can be used for arbitrary d > 2 (see Górecki and Holeňa (2013) for more details including an example for d = 4). It returns the sets U C (ψ k ), k = 1, ..., d − 1.…”

Section: Our Approachmentioning

confidence: 99%

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

2015

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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.

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“…The literature on structure learning for these models is emerging. For example, algorithms have begun to be developed for hierarchical Archimedean copulas, see, e.g., Okhrin et al (2013); Segers and Uyttendaele (2014); Górecki et al (2016Górecki et al ( , 2017; Cossette et al (2019) and references therein. Learning the block structure of a matrix of pair-wise Kendall's tau under the partial exchangeability of the copula has been considered by Perreault et al (2019).…”

Section: Introductionmentioning

confidence: 99%

Hypothesis tests for structured rank correlation matrices

Perreault,

Neslehova,

Duchesne

2020

Preprint

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Joint modeling of a large number of variables often requires dimension reduction strategies that lead to structural assumptions of the underlying correlation matrix, such as equal pair-wise correlations within subsets of variables. The underlying correlation matrix is thus of interest for both model specification and model validation. In this paper, we develop tests of the hypothesis that the entries of the Kendall rank correlation matrix are linear combinations of a smaller number of parameters. The asymptotic behaviour of the proposed test statistics is investigated both when the dimension is fixed and when it grows with the sample size. We pay special attention to the restricted hypothesis of partial exchangeability, which contains full exchangeability as a special case. We show that under partial exchangeability, the test statistics and their large-sample distributions simplify, which leads to computational advantages and better performance of the tests. We propose various scalable numerical strategies for implementation of the proposed procedures, investigate their finite sample behaviour through simulations, and demonstrate their use on a real dataset of mean sea levels at various geographical locations.

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

Cited by 13 publications

References 19 publications

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

Outer power transformations of hierarchical Archimedean copulas: Construction, sampling and estimation

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

Hypothesis tests for structured rank correlation matrices

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