Estimating axial symmetry using random projections

Abstract

We study the directions of axial symmetry of multivariate distributions and relate the measure or cardinality of this set to spherical symmetry. Under Carleman's condition, two independent random projections identify all true axes of symmetry in \(\RR^2\) almost surely. We propose a level-set estimator of the symmetry axes and prove its Hausdorff consistency, in arbitrary dimension, for the zero set of the population criterion based on finitely many projections. Together with planar identification, this gives a consistent estimation of the true axes of symmetry in dimension two.

arXiv comment29 pages, 8 figures, 5 tables