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 few hitherto proposed tests only a class of tests studied in Henze et al. (J Multivar Anal 87:275–297, 2003 ) that depends on a positive parameter a respects this property. We identify a measure of deviation $$\varDelta _a$$ Δ a (say) from symmetry associated with the test statistic $$T_{n,a}$$ T n , a (say), and we obtain the limit normal distribution of $$T_{n,a}$$ T n , a as $$n \rightarrow \infty $$ n → ∞ 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 $$\varDelta _a$$ Δ a . 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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有关未知中心的多元反射对称性的 HKM 检验的更多好消息

我们重新审视关于未指定点的多元反射对称性的测试问题。尽管这个测试问题对于全秩仿射变换是不变的，但在迄今为止提出的为数不多的测试中，只有 Henze 等人研究的一类测试。(J Multivar Anal 87:275–297, 2003 ) 依赖于正参数 a 尊重这个属性。我们从与检验统计量 $$T_{n,a}$$ T n , a (say) 相关联的对称性中确定偏差 $$\varDelta _a$$ Δ a (say)，并且我们获得极限正态分布$$T_{n,a}$$ T n , a as $$n \rightarrow \infty $$ n → ∞ 在对称性的固定替代方案下。由于此极限正态分布的方差的一致估计量可用，因此我们获得了 $$\varDelta _a$$ Δ a 的渐近置信区间。考试，