A Necessary Test of Goodness of Fit for Sphericity

A p × 1 random vector x is said to have a spherical distribution, if for every p × p orthogonal matrix Q, x and Qx have the same distribution. In this paper, some nonparametric goodness of fit Wilcoxon-type tests for sphericity are proposed. Some results on the limiting distributions of the statistics are obtained. These results provide a theoretical basis for a necessary test of sphericity.