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.

中文翻译：

---

球对称和椭圆对称测试

我们基于具有有限均值的随机向量 $X$ 具有球面分布当且仅当 $\Ex[u^\top X | v^\top X] = 0$ 对任意两个垂直向量 $u$ 和 $v$ 成立。我们的测试基于 Kolmogorov-Smirnov 统计量，其拒绝域是通过球对称自举法找到的。我们使用具有某种独立意义的一般 Donsker 定理来展示球对称自举测试的一致性。对于椭圆对称性检验的情况，由于估计的位置和尺度参数，Kolmogorov-Smirnov 统计量具有渐近漂移项。因此，在引导程序中需要额外的标准化。在模拟研究中，