Testing for spherical and elliptical symmetry
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We construct new testing procedures for spherical and elliptical symmetry based on the characterization that a random vector X with finite mean has a spherical distribution if and only if [u^ X | v^ X] = 0 holds for any two perpendicular vectors u and v. Our test is based on the Kolmogorov-Smirnov statistic, and its rejection region is found via the spherically symmetric bootstrap. We show the consistency of the spherically symmetric bootstrap test using a general Donsker theorem which is of some independent interest. For the case of testing for elliptical symmetry, the Kolmogorov-Smirnov statistic has an asymptotic drift term due to the estimated location and scale parameters. Therefore, an additional standardization is required in the bootstrap procedure. In a simulation study, the size and the power properties of our tests are assessed for several distributions and the performance is compared to that of several competing procedures.
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