Improved distance correlation estimation

Abstract

[Abstract]: Distance correlation is a novel class of multivariate dependence measure, taking positive values between 0 and 1, and applicable to random vectors of not necessarily equal arbitrary dimensions. It offers several advantages over the well-known Pearson correlation coefficient, the most important being that distance correlation equals zero if-and-only if- the random vectors are independent. There are two different estimators of the distance correlation available in the literature. The first estimator, proposed by Székely et al. (Ann Stat 35:2769–279 2007), is based on an asymptotically unbiased estimator of the distance covariance, which is a V-statistic. The second builds on an unbiased estimator of the distance covariance proposed in Székely and Rizzo (Stat 42:2382–2412 2014), shown to be a U-statistic by Huo and Székely (Technometrics 58:435–447 2016). This study evaluates their efficiency (mean squared error) and compares computational times for both methods under different dependence structures. Under conditions of independence or near-independence, the V-estimates are biased, while the U-estimator frequently cannot be computed due to negative values. To address this challenge, a convex linear combination of the former estimators is proposed and studied, yielding good results regardless of the level of dependence. Additionally, a medical database is studied and discussed.