Smooth test for elliptical symmetry

In this paper, we suggest the smooth test procedures for testing the hypothesis of elliptical symmetry. The test statistic is constructed by using the Cholesky decomposition and spherical harmonics. The asymptotic null distributions of the transformed samples are obtained, and a bootstrapping algorithm is given to approximate the critical values of the test statistic for a wide range of sample sizes. We also introduce the method for testing the multivariate distributions within the subclasses of elliptically symmetric distributions. An example is given to illustrate the goodness-of-fit test for multivariate t-distributions.