More good news on the HKM test for multivariate reflected symmetry about an unknown centre

  • Norbert HenzeEmail author
  • Celeste Mayer


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.


Test for reflected symmetry Fixed alternatives Affine invariance Weighted \(L^2\)-statistic Elliptical symmetry 



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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Copyright information

© The Institute of Statistical Mathematics, Tokyo 2019

Authors and Affiliations

  1. 1.Institute of StochasticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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