Nonparametric tests of independence between random vectors

Authors:
R. Beran;M. Bilodeau;P. Lafaye de Micheaux
Affiliations:
UC Davis Department of Statistics, 360 Kerr Hall, One Shields Ave., Davis, CA 95616, USA;Département de mathématiques et de statistique, université de Montréal, C.P. 6128, succursale Centre-ville, Montréal, Canada H3C 3J7;Université Pierre Mendès France, Laboratoire de Statistique et Analyse de Données (LabSAD), BP 47/F-38040 Grenoble Cedex 9, France
Venue:
Journal of Multivariate Analysis
Year:
2007

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Abstract

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 Mobius. 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 R"n","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.