High Dimensional Rank Tests for Sphericity

Sphericity test plays a key role in many statistical problems. We propose Spearman's rho-type rank test and Kendall's tau-type rank test for sphericity in the high dimensional settings. We show that these two tests are equivalent. Thanks to the "blessing of dimension", we do not need to estimate any nuisance parameters. Without estimating the location parameter, we can allow the dimension to be arbitrary large. Asymptotic normality of these two tests are also established under elliptical distributions. Simulations demonstrate that they are very robust and efficient in a wide range of settings.