In a seminal paper, Sejdinovic et al. (Ann. Statist. 41 (2013) 2263-2291) showed the equivalence of the Hilbert-Schmidt Independence Criterion (HSIC) and a generalization of distance covariance. In... Show moreIn a seminal paper, Sejdinovic et al. (Ann. Statist. 41 (2013) 2263-2291) showed the equivalence of the Hilbert-Schmidt Independence Criterion (HSIC) and a generalization of distance covariance. In this paper, the two notions of dependence are unified with a third prominent concept for independence testing, the "global test" introduced in (J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 (2006) 477-493). The new viewpoint provides novel insights into all three test traditions, as well as a unified overall view of the way all three tests contrast with classical association tests. As our main result, a regression perspective on HSIC and generalized distance covariance is obtained, allowing such tests to be used with nuisance covariates or for survival data. Several more examples of cross-fertilization of the three traditions are provided, involving theoretical results and novel methodology. To illustrate the difference between classical statistical tests and the unified HSIC/distance covariance/global tests we investigate the case of association between two categorical variables in depth. Show less
