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On the specification of multivariate association measures and their behaviour with increasing dimension

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  • Gijbels, Irène
  • Kika, Vojtěch
  • Omelka, Marek

Abstract

In this paper the interest is to elaborate on the generalization of bivariate association measures, namely Spearman’s rho, Kendall’s tau, Blomqvist’s beta and Gini’s gamma, for a general dimension d≥2. Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii) a conical combination of all components; (iii) a set of arbitrary random components. Existing generalizations are evaluated with respect to the axiom set. For a d-variate Gini’s gamma, a simplified formula is developed, making its analytical computation easier. Further, for Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension d increases is studied. Nonparametric estimation of the considered generalizations of multivariate association measures is reviewed and a nonparametric estimator of the multivariate Gini’s gamma is introduced. The practical use of multivariate association measures is illustrated on a real data example.

Suggested Citation

  • Gijbels, Irène & Kika, Vojtěch & Omelka, Marek, 2021. "On the specification of multivariate association measures and their behaviour with increasing dimension," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    • Handle: RePEc:eee:jmvana:v:182:y:2021:i:c:s0047259x20302852
    DOI: 10.1016/j.jmva.2020.104704
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    References listed on IDEAS

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    Cited by:

    1. Fuchs, Sebastian & Di Lascio, F. Marta L. & Durante, Fabrizio, 2021. "Dissimilarity functions for rank-invariant hierarchical clustering of continuous variables," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).
    2. Aleksy Leeuwenkamp & Wentao Hu, 2023. "New general dependence measures: construction, estimation and application to high-frequency stock returns," Papers 2309.00025, arXiv.org.
    3. Blier-Wong, Christopher & Cossette, Hélène & Marceau, Etienne, 2022. "Stochastic representation of FGM copulas using multivariate Bernoulli random variables," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    4. Gijbels Irène & Matterne Margot, 2021. "Study of partial and average conditional Kendall’s tau," Dependence Modeling, De Gruyter, vol. 9(1), pages 82-120, January.

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