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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.

##### MSC:
 62H11 Directional data; spatial statistics 62H15 Hypothesis testing in multivariate analysis 62F05 Asymptotic properties of parametric tests
##### Software:
CircStats; circular
Full Text:
##### References:
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