Testing for Spherical Symmetry of a Multivariate Distribution

We consider a test for spherical symmetry of a distribution in Rdwith an unknown center. It is a multivariate version of the tests suggested by Schuster and Barker and by Arcones and Giné. The test statistic is based on the multivariate extension of the distribution and quantile functions, recently introduced by Koltchinskii and Dudley and by Chaudhuri. We study the asymptotic behavior of the sequence of test statistics for large samples and for a fixed spherically asymmetric alternative as well as for a sequence of local alternatives converging to a spherically symmetric distribution. We also study numerically the performance of the test for moderate sample sizes and justify a symmetrized version of bootstrap approximation of the distribution of test statistics.