Testing Group Symmetry of a Multivariate Distribution

Abstract: We propose and study a general class of tests for group symmetry of a multivariate distribution, which encompasses different types of symmetry, such as ellipsoidal and permutation symmetries among others. Our approach is based on supremum norms of special empirical processes combined with bootstrap. We show that these tests are consistent against any fixed alternative. This work generalizes the methodology of Koltchinskii and Sakhanenko [7], developed for ellipsoidal symmetry to the case of group symmetry. It also provides a unified approach to testing different types of symmetry of a multivariate distribution.