关于（唯一的）仿射不变测试的更多好消息反映了关于未知中心的对称性
More good news on the (only) affine invariant test for multivariate reflected symmetry about an unknown center
摘要：
我们重新审视的问题，测试多元反射对称性的一个未指定的点。虽然这个测试问题对于满秩仿射变换是不变的，但在迄今为止提出的几个测试中，只有[12]中研究的测试尊重这个属性。我们确定了一个测量偏差$Δ$（比方说）与对称性相关的检验统计量$T_n$（比方说），我们得到了极限正态分布的$T_n$为$n \to \infty$下一个固定的替代对称性。由于这个极限正态分布的方差的相合估计量是可用的，我们得到了$Δ$的渐近置信区间。该测试，当应用到一个经典的数据集，强烈拒绝反射对称的假设，虽然其他测试甚至不反对更强的假设椭圆对称。
英文摘要：
We revisit the problem of testing for multivariate reflected symmetry about an unspecified point. Although this testing problem is invariant with respect to full-rank affine transformations, among the hitherto few proposed tests only the test studied in [12] respects this property. We identify a measure of deviation $Δ$ (say) from symmetry associated with the test statistic $T_n$ (say), and we obtain the limit normal distribution of $T_n$ as $n \to \infty$ under a fixed alternative to symmetry. Since a consistent estimator of the variance of this limit normal distribution is available, we obtain an asymptotic confidence interval for $Δ$. The test, when applied to a classical data set, strongly rejects the hypothesis of reflected symmetry, although other tests even do not object against the much stronger hypothesis of elliptical symmetry.
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