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arXiv · 2004.13151

Testing for spherical and elliptical symmetry

Isaia Albisetti·Fadoua Balabdaoui·Hajo Holzmann
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Abstract

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 $\Ex[u^\top X | v^\top X] = 0$ 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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Isaia Albisetti, Fadoua Balabdaoui, Hajo Holzmann. 2020-04-27. Testing for spherical and elliptical symmetry. https://arxiv.org/abs/2004.13151

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