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On the uniqueness of distance covariance

Author

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  • Székely, Gábor J.
  • Rizzo, Maria L.

Abstract

Distance covariance and distance correlation are non-negative real numbers that characterize the independence of random vectors in arbitrary dimensions. In this work we prove that distance covariance is unique, starting from a definition of a covariance as a weighted L2 norm that measures the distance between the joint characteristic function of two random vectors and the product of their marginal characteristic functions. Rigid motion invariance and scale equivariance of these weighted L2 norms imply that the weight function of distance covariance is unique.

Suggested Citation

  • Székely, Gábor J. & Rizzo, Maria L., 2012. "On the uniqueness of distance covariance," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2278-2282.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:12:p:2278-2282
    DOI: 10.1016/j.spl.2012.08.007
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    Citations

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    Cited by:

    1. Shang, Du & Shang, Pengjian, 2022. "The dependence measurements based on martingale difference correlation and distance correlation: Efficient tools to distinguish different complex systems," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    2. Górecki Tomasz & Krzyśko Mirosław & Ratajczak Waldemar & Wołyński Waldemar, 2016. "An Extension of the Classical Distance Correlation Coefficient for Multivariate Functional Data with Applications," Statistics in Transition New Series, Statistics Poland, vol. 17(3), pages 449-466, September.
    3. Tomasz Górecki & Mirosław Krzy´sko & Waldemar Ratajczak & Waldemar Woły´nski, 2016. "An Extension Of The Classical Distance Correlation Coefficient For Multivariate Functional Data With Applications," Statistics in Transition New Series, Polish Statistical Association, vol. 17(3), pages 449-466, September.
    4. Fortuna, Francesca & Naccarato, Alessia & Salvati, Luca, 2024. "The functional distance-based approach: An application on long-term Metropolitan Development," Socio-Economic Planning Sciences, Elsevier, vol. 94(C).
    5. Mirosław Krzyśko & Tomasz Górecki & Waldemar Wołyński & Waldemar Ratajczak, 2016. "An Extension of the Classical Distance Correlation Coefficient for Multivariate Functional Data with Applications," Statistics in Transition new series, Główny Urząd Statystyczny (Polska), vol. 17(3), pages 449-466, September.
    6. Tomasz Górecki & Mirosław Krzyśko & Waldemar Wołyński, 2019. "Variable Selection In Multivariate Functional Data Classification," Statistics in Transition New Series, Polish Statistical Association, vol. 20(2), pages 123-138, June.
    7. Górecki Tomasz & Krzyśko Mirosław & ński Waldemar Woły, 2019. "Variable Selection In Multivariate Functional Data Classification," Statistics in Transition New Series, Statistics Poland, vol. 20(2), pages 123-138, June.
    8. Fan, Yanan & de Micheaux, Pierre Lafaye & Penev, Spiridon & Salopek, Donna, 2017. "Multivariate nonparametric test of independence," Journal of Multivariate Analysis, Elsevier, vol. 153(C), pages 189-210.
    9. Dueck, Johannes & Edelmann, Dominic & Richards, Donald, 2017. "Distance correlation coefficients for Lancaster distributions," Journal of Multivariate Analysis, Elsevier, vol. 154(C), pages 19-39.

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