The tests are constructed from the Mobius transformation applied to the probability cells in a multi-way contingency table. The Pearson chi-squared test of mutual independence is partitioned into A-dependence statistics over all subsets A of variables. The goal of the partition is to identify subsets of dependent variables when the mutual independence hypothesis is rejected by the Pearson chi-squared test. The methodology can be directly adapted to test for serial independence of d successive observations of a stationary categorical time series.
This function can be used for the following two problems: 1) testing mutual independence between many numerical random vectors, and 2) testing for serial independence of a multivariate stationary numerical time series. The proposed test does not assume continuous marginals. It is valid for any probability distribution. It is also invariant with respect to the affine general linear group of transformations on the vectors. 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. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence can not be distribution free. The critical values of the proposed test are thus computed with the bootstrap which was shown to be consistent in this context.
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