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Skew-rotationally-symmetric distributions and related efficient inferential procedures. (English) Zbl 1368.62134
Summary: Most commonly used distributions on the unit hypersphere \(\mathcal{S}^{k - 1} = \{\mathbf{v} \in \mathbb{R}^k : \mathbf{v}^\top \mathbf{v} = 1 \}\), \(k \geq 2\), assume that the data are rotationally symmetric about some direction \(\mathbf{\theta} \in \mathcal{S}^{k - 1}\). 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.

62H11 Directional data; spatial statistics
62H15 Hypothesis testing in multivariate analysis
62F05 Asymptotic properties of parametric tests
CircStats; circular
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