Measuring deviations from spherical symmetry

Abstract

Most of the work on checking spherical symmetry assumptions on the distribution of the $p$-dimensional random vector $Y$ has its focus on statistical tests for the null hypothesis of exact spherical symmetry. In this paper, we take a different point of view and propose a measure for the deviation from spherical symmetry, which is based on the minimum distance between the distribution of the vector $(\Vert Y\Vert ,Y/\Vert Y\Vert {)}^{\top }$ and its best approximation by a distribution of a vector $(\Vert Y_s\Vert ,Y_s/\Vert Y_s\Vert {)}^{\top }$ corresponding to a random vector $Y_s$ with a spherical distribution. We develop estimators for the minimum distance with corresponding statistical guarantees (provided by asymptotic theory) and demonstrate the applicability of our approach by means of a simulation study and a real data example.