More good news on the (only) affine invariant test for multivariate reflected symmetry about an unknown center

TL;DR

This paper revisits an affine invariant test for multivariate reflected symmetry, providing a new measure of deviation, deriving its asymptotic distribution, and demonstrating its effectiveness through data analysis.

Contribution

It introduces a new deviation measure and derives the asymptotic distribution of the test statistic for the affine invariant symmetry test.

Findings

The test strongly rejects symmetry in a classical dataset.

The test outperforms others in detecting asymmetry.

An asymptotic confidence interval for the deviation measure is established.

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 hitherto few proposed tests only the test studied in [12] respects this property. We identify a measure of deviation $\Delta$ (say) from symmetry associated with the test statistic $T_n$ (say), and we obtain the limit normal distribution of $T_n$ as $n \to \infty$ 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 $\Delta$. 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.