Skew-rotationally-symmetric distributions and related efficient inferential procedures

Abstract

Most commonly used distributions on the unit hypersphere ${\mathcal{S}}^{k-1}=\{\mathbf{v}\in {\mathbb{R}}^k:{\mathbf{v}}^{\mathrm{\top }}\mathbf{v}=1\}$, $k\ge 2$, assume that the data are rotationally symmetric about some direction $\mathit{\theta }$ is an element of ${\mathcal{S}}^{k-1}$. However, there is empirical evidence that this assumption often fails to describe reality. We study in this paper a new class of skew-rotationally-symmetric distributions on ${\mathcal{S}}^{k-1}$ that enjoy numerous good properties. We discuss the Fisher information structure of the model and derive efficient inferential procedures. In particular, we obtain the first semi-parametric test for rotational symmetry about a known direction. We also propose a second test for rotational symmetry, obtained through the definition of a new measure of skewness on the hypersphere. We investigate the finite-sample behavior of the new tests through a Monte Carlo simulation study. We conclude the paper with a discussion about some intriguing open questions related to our new models.