The distance correlation t-test of independence in high dimension

AMS subject classifications: primary 62G10 secondary 62H20 Keywords: dCor dCov Multivariate independence Distance covariance Distance correlation High dimension a b s t r a c t Distance correlation is extended to the problem of testing the independence of random vectors in high dimension. Distance correlation characterizes independence and determines a test of multivariate independence for random vectors in arbitrary dimension. In this work, a modified distance correlation statistic is proposed, such that under independence the distribution of a transformation of the statistic converges to Student t, as dimension tends to infinity. Thus we obtain a distance correlation t-test for independence of random vectors in arbitrarily high dimension, applicable under standard conditions on the coordinates that ensure the validity of certain limit theorems. This new test is based on an unbiased es-timator of distance covariance, and the resulting t-test is unbiased for every sample size greater than three and all significance levels. The transformed statistic is approximately normal under independence for sample size greater than nine, providing an informative sample coefficient that is easily interpretable for high dimensional data. 1. Introduction Many applications in genomics, medicine, engineering, etc. require analysis of high dimensional data. Time series data can also be viewed as high dimensional data. Objects can be represented by their characteristics or features as vectors p. In this work, we consider the extension of distance correlation to the problem of testing independence of random vectors in arbitrarily high, not necessarily equal dimensions, so the dimension p of the feature space of a random vector is typically large. measure all types of dependence between random vectors in arbitrary, not necessarily equal dimensions. (See Section 2 for definitions.) Distance correlation takes values in [0, 1] and is equal to zero if and only if independence holds. It is more general than the classical Pearson product moment correlation, providing a scalar measure of multivariate independence that characterizes independence of random vectors. The distance covariance test of independence is consistent against all dependent alternatives with finite second moments. In practice, however, researchers are often interested in interpreting the numerical value of distance correlation, without a formal test. For example, given an array of distance correlation statistics, what can one learn about the strength of dependence relations from the dCor statistics without a formal test? This is in fact, a difficult question, but a solution is finally available for a large class of problems. The …