论文标题
测试球形和椭圆形对称性
Testing for spherical and elliptical symmetry
论文作者
论文摘要
我们基于以下特征来为球形和椭圆形对称性构建新的测试程序，即随机向量$ x $具有有限的均值时，只有$ \ ex [u^\ ex [u^\ top x | v^\ top x] = 0 $保留任何两个垂直向量$ u $和$ v $。我们的测试基于Kolmogorov-Smirnov统计量，其拒绝区域是通过球形对称的引导程序找到的。我们使用普通的Donsker定理显示了球形对称性引导测试的一致性，该定理具有独立的兴趣。对于测试椭圆形对称性的情况，由于估计的位置和比例参数，Kolmogorov-Smirnov统计量具有渐近漂移项。因此，在引导程序中需要附加标准化。在一项仿真研究中，评估了我们的测试的大小和功率特性，并将其与几个竞争程序的性能进行比较。
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.
