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Nonparametric tests of independence between random vectors


  • Beran, R.
  • Bilodeau, M.
  • Lafaye de Micheaux, P.


A nonparametric test of the mutual independence between many numerical random vectors is proposed. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Möbius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. If the number of vectors is p and there are n observations, the test is defined from a collection of processes Rn,A, where A is a subset of {1,...,p} of cardinality A>1, which are asymptotically independent and Gaussian. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence cannot be distribution free. The critical values of the proposed test are thus computed with the bootstrap which is shown to be consistent. Another similar test, with the same asymptotic properties, for the serial independence of a multivariate stationary sequence is also proposed. The proposed test works when some or all of the marginal distributions are singular with respect to Lebesgue measure. Moreover, in singular cases described in Section 4, the test inherits useful invariance properties from the general affine invariance property.

Suggested Citation

  • Beran, R. & Bilodeau, M. & Lafaye de Micheaux, P., 2007. "Nonparametric tests of independence between random vectors," Journal of Multivariate Analysis, Elsevier, vol. 98(9), pages 1805-1824, October.
  • Handle: RePEc:eee:jmvana:v:98:y:2007:i:9:p:1805-1824

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    References listed on IDEAS

    1. Deheuvels, Paul, 1981. "An asymptotic decomposition for multivariate distribution-free tests of independence," Journal of Multivariate Analysis, Elsevier, vol. 11(1), pages 102-113, March.
    2. Bilodeau, M. & Lafaye de Micheaux, P., 2005. "A multivariate empirical characteristic function test of independence with normal marginals," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 345-369, August.
    3. Ghoudi, Kilani & Kulperger, Reg J. & Rémillard, Bruno, 2001. "A Nonparametric Test of Serial Independence for Time Series and Residuals," Journal of Multivariate Analysis, Elsevier, vol. 79(2), pages 191-218, November.
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    Cited by:

    1. Kojadinovic, Ivan & Holmes, Mark, 2009. "Tests of independence among continuous random vectors based on Cramr-von Mises functionals of the empirical copula process," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1137-1154, July.
    2. Ivan Kojadinovic & Jun Yan, 2011. "Tests of serial independence for continuous multivariate time series based on a Möbius decomposition of the independence empirical copula process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(2), pages 347-373, April.
    3. repec:eee:jmvana:v:168:y:2018:i:c:p:304-322 is not listed on IDEAS
    4. 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.
    5. Györfi, László & Walk, Harro, 2012. "Strongly consistent nonparametric tests of conditional independence," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1145-1150.
    6. Kojadinovic, Ivan, 2010. "Hierarchical clustering of continuous variables based on the empirical copula process and permutation linkages," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 90-108, January.


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