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Four simple axioms of dependence measures

Author

Listed:
  • Tamás F. Móri

    (ELTE Eötvös Loránd University)

  • Gábor J. Székely

    (National Science Foundation
    Hungarian Academy of Sciences)

Abstract

Recently new methods for measuring and testing dependence have appeared in the literature. One way to evaluate and compare these measures with each other and with classical ones is to consider what are reasonable and natural axioms that should hold for any measure of dependence. We propose four natural axioms for dependence measures and establish which axioms hold or fail to hold for several widely applied methods. All of the proposed axioms are satisfied by distance correlation. We prove that if a dependence measure is defined for all bounded nonconstant real valued random variables and is invariant with respect to all one-to-one measurable transformations of the real line, then the dependence measure cannot be weakly continuous. This implies that the classical maximal correlation cannot be continuous and thus its application is problematic. The recently introduced maximal information coefficient has the same disadvantage. The lack of weak continuity means that as the sample size increases the empirical values of a dependence measure do not necessarily converge to the population value.

Suggested Citation

  • Tamás F. Móri & Gábor J. Székely, 2019. "Four simple axioms of dependence measures," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(1), pages 1-16, January.
    • Handle: RePEc:spr:metrik:v:82:y:2019:i:1:d:10.1007_s00184-018-0670-3
    DOI: 10.1007/s00184-018-0670-3
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    References listed on IDEAS

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    1. Huang, Qiming & Zhu, Yu, 2016. "Model-free sure screening via maximum correlation," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 89-106.
    2. Papadatos, Nickos & Xifara, Tatiana, 2013. "A simple method for obtaining the maximal correlation coefficient and related characterizations," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 102-114.
    3. Sampson, Allan R., 1984. "A multivariate correlation ratio," Statistics & Probability Letters, Elsevier, vol. 2(2), pages 77-81, March.
    4. Papadatos, Nickos, 2014. "Some counterexamples concerning maximal correlation and linear regression," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 114-117.
    Full references (including those not matched with items on IDEAS)

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