Testing axial symmetry around an unspecified direction

TL;DR

This paper develops a statistical test for axial symmetry in multivariate distributions without knowing the symmetry axis, using projected data and bootstrap methods.

Contribution

It introduces a new testing procedure that reduces the problem to finite candidate directions and proves its asymptotic and bootstrap validity.

Findings

The test effectively identifies axial symmetry in multivariate data.

Asymptotic distribution of the test statistic is derived.

Bootstrap method provides valid inference under regularity conditions.

Abstract

We consider the problem of testing whether a multivariate distribution is axially symmetric about some unknown direction. Under a simple-spectrum assumption on the covariance matrix, any symmetry axis must coincide with an eigenvector of the covariance matrix, so the problem reduces to testing a finite set of candidate directions. For each candidate direction, we construct a Kolmogorov--Smirnov-type statistic based on projected data and sample splitting. We derive its asymptotic distribution in a triangular-array framework and establish bootstrap validity under suitable regularity conditions. This leads to a feasible testing procedure for axial symmetry when the symmetry direction is unspecified.