A necessary test for elliptical symmetry based on the uniform distribution over the Stiefel manifold

Abstract

This paper provides a new procedure for testing the null hypothesis of multivariate elliptical symmetry. A test for uniformity over the Stiefel manifold based on modified degenerate $V$-statistics is employed since the test statistic proposed in this paper consists of independent random matrices, formed by the scaled residuals (or the Studentized residuals), which are uniformly distributed over the Stiefel manifold under the null hypothesis. Also, Monte Carlo simulation studies are carried out to evaluate the type I error and power of the test. Finally, the procedure is illustrated using the Iris data.