Reducing multivariate independence testing to two bivariate means comparisons
arxiv(2024)
摘要
Testing for independence between two random vectors is a fundamental problem
in statistics. It is observed from empirical studies that many existing omnibus
consistent tests may not work well for some strongly nonmonotonic and nonlinear
relationships. To explore the reasons behind this issue, we novelly transform
the multivariate independence testing problem equivalently into checking the
equality of two bivariate means. An important observation we made is that the
power loss is mainly due to cancellation of positive and negative terms in
dependence metrics, making them very close to zero. Motivated by this
observation, we propose a class of consistent metrics with a positive integer
γ that exactly characterize independence. Theoretically, we show that
the metrics with even and infinity γ can effectively avoid the
cancellation, and have high powers under the alternatives that two mean
differences offset each other. Since we target at a wide range of dependence
scenarios in practice, we further suggest to combine the p-values of test
statistics with different γ's through the Fisher's method. We illustrate
the advantages of our proposed tests through extensive numerical studies.
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