On some consistent tests of mutual independence among several random vectors of arbitrary dimensions
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On some consistent tests of mutual independence among several random vectors of arbitrary dimensions Angshuman Roy1
· Soham Sarkar2
· Anil K. Ghosh3
· Alok Goswami3
Received: 22 August 2019 / Accepted: 29 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Testing for mutual independence among several random vectors is a challenging problem, and in recent years, it has gained significant attention in statistics and machine learning literature. Most of the existing tests of independence deal with only two random vectors, and they do not have straightforward generalizations for testing mutual independence among more than two random vectors of arbitrary dimensions. On the other hand, there are various tests for mutual independence among several random variables, but these univariate tests do not have natural multivariate extensions. In this article, we propose two general recipes, one based on inter-point distances and the other based on linear projections, for multivariate extensions of these univariate tests. Under appropriate regularity conditions, these resulting tests turn out to be consistent whenever we have consistency for the corresponding univariate tests. We carry out extensive numerical studies to compare the empirical performance of these proposed methods with the state-of-the-art methods. Keywords Copula distribution · Cramér–Wold device · Inter-point distance · Maximum mean discrepancy · Multi-scale approach · Permutation test
1 Introduction Suppose that there are n independent observations on a d-dimensional random vector X = (X(1) , . . . , X( p) ) ∼ F, and we want to test whether the sub-vectors X(1) ∈ Rd1 , . . . , X( p) ∈ Rd p (d1 + . . . + d p = d) are mutually independent. This is one of the fundamental and challenging problems in statistics and machine learning, and it has
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Angshuman Roy [email protected] Soham Sarkar [email protected] Anil K. Ghosh [email protected] Alok Goswami [email protected]
1
Applied Statistics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
2
Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Station 8, 1015 Lausanne, Switzerland
3
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
recently gained considerable interest among the researchers in these communities. For p = 2, this problem has been extensively investigated by many researchers, and some tests have been developed for it. The dCov test based on distance correlation (Székely et al. 2007), the HSIC test based on Hilbert–Schmidt independence criterion (see, e.g., Gretton et al. 2007; Gretton and Gyorfi 2010) and the test based on 2 × 2 contingency tables formed using ranks of distances (Heller et al. 2013) are consistent under general alternatives. Sejdinovic et al. (2013) showed some equivalence between the dCov test and the HSIC test. Other tests of independence between two random vectors include the graph-based tests proposed by Friedm
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