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

Norbert Henze () and Celeste Mayer
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Norbert Henze: Karlsruhe Institute of Technology (KIT)
Celeste Mayer: Karlsruhe Institute of Technology (KIT)

Annals of the Institute of Statistical Mathematics, 2020, vol. 72, issue 3, No 5, 770 pages

Abstract: 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 $$\varDelta _a$$Δa (say) from symmetry associated with the test statistic $$T_{n,a}$$Tn,a (say), and we obtain the limit normal distribution of $$T_{n,a}$$Tn,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 $$\varDelta _a$$Δa. 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.

Keywords: Test for reflected symmetry; Fixed alternatives; Affine invariance; Weighted $$L^2$$ L 2 -statistic; Elliptical symmetry (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10463-019-00707-5

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