Measuring deviations from spherical symmetry
Lujia Bai, Holger Dette
[stat.ME,stat.TH]
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 $\big (|Y|, Y/ |Y| )^\top $ and its best approximation by a distribution of a vector $\big (|Y_s|, Y_s/ |Y_s | )^\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.