A consistent test of spherical symmetry for multivariate and high-dimensional data via data augmentation

We develop a test for spherical symmetry of a multivariate distribution P that works even when the dimension of the data d is larger than the sample size n. We propose a non-negative measure (P) such that (P) =0 if and only if P is spherically symmetric. We construct a consistent estimator of (P) using the data augmentation method and investigate its large sample properties. The proposed test based on this estimator is calibrated using a novel resampling algorithm. Our test controls the Type-I error, and it is consistent against general alternatives. We also study its behaviour for a sequence of alternatives (1-ₙ) F+ₙ G, where (G) =0 but (F) >0, and ₙ 0, 1. When ₙ0. Moreover, our test is provably consistent for high-dimensional data even when d is larger than n. Our numerical results amply demonstrate the superiority of the proposed test over some state-of-the-art methods.