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

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 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 Delta a (say) from symmetry associated with the test statistic Tn,a (say), and we obtain the limit normal distribution of Tn,a as n ->infinity 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 Delta 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.