The distance correlation t-test of independence in high dimension

• Published in: Journal of multivariate analysis 2013-05, Vol.117, p.193-213
• Main Authors: Székely, Gábor J., Rizzo, Maria L.
• Format: Article
• Language: English
• Subjects: Algebra; Correlation analysis; dCor; dCov; Distance correlation; Distance covariance; High dimension; Independence; Multivariate independence; Probability distribution; Sample size; Studies
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2013.02.012

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 estimator 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.