Abstract: Three important projection-pursuit correlations, namely, the distance, projection, and multivariate Blum-Kiefer-Rosenblatt (BKR) correlations, have been proposed in the literature to test for independence between two random vectors in arbitrary dimensions. In this study, we compare the asymptotic power performance of independence tests built upon these three projection-pursuit correlations, in a uniform sense. We show that in the presence of outliers, the projection and multivariate BKR correlation tests are still powerful, whereas the distance correlation test may lose power. We also analyze the minimax optimality of these independence tests. We show that their minimum separation rates are of order 𝓃^–1, where 𝓃 stands for the sample size, and that this minimax optimal rate is tight in terms of the projection, distance, and multivariate BKR correlations.

Key words and phrases: Distance correlation, independence test, minimax optimality, projection correlation, power function, robustness.