Abstract
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 \(\varDelta _a\) (say) from symmetry associated with the test statistic \(T_{n,a}\) (say), and we obtain the limit normal distribution of \(T_{n,a}\) as \(n \rightarrow \infty \) 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 \(\varDelta _a\). 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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The authors wish to thank two anonymous referees and a member of the Editorial Board for their careful reading of the manuscript and for many helpful suggestions.
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Henze, N., Mayer, C. More good news on the HKM test for multivariate reflected symmetry about an unknown centre. Ann Inst Stat Math 72, 741–770 (2020). https://doi.org/10.1007/s10463-019-00707-5
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DOI: https://doi.org/10.1007/s10463-019-00707-5