Rotationally invariant tests based on test statistics of the von Mises type are proposed under the hypothesis of spherical symmetry of a multivariate distribution. The tests are distribution-free when the hypothesis of spherical symmetry is true. The asymptotic distribution of the test statistics are derived under the null hypothesis and under any fixed alternative. A simple criterion for consistency is given. The results are illustrated by numerous examples of test statistics which give rise to tests being consistent against all alternatives.
"Testing for Spherical Symmetry of a Multivariate Distribution." Ann. Statist. 19 (2) 899 - 917, June, 1991. https://doi.org/10.1214/aos/1176348127