A Regression Perspective on Generalized Distance Covariance and the Hilbert–Schmidt Independence Criterion

In a seminal paper, Sejdinovic, et al. [49] showed the equivalence of the Hilbert-Schmidt Independence Criterion (HSIC) [20] and a generalization of distance covariance [62]. In this paper the two notions of dependence are unified with a third prominent concept for independence testing, the “global test” introduced in [16]. 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.