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2020 On the power of axial tests of uniformity on spheres
Christine Cutting, Davy Paindaveine, Thomas Verdebout
Electron. J. Statist. 14(1): 2123-2154 (2020). DOI: 10.1214/20-EJS1716

Abstract

Testing uniformity on the $p$-dimensional unit sphere is arguably the most fundamental problem in directional statistics. In this paper, we consider this problem in the framework of axial data, that is, under the assumption that the $n$ observations at hand are randomly drawn from a distribution that charges antipodal regions equally. More precisely, we focus on axial, rotationally symmetric, alternatives and first address the problem under which the direction $\boldsymbol{\theta}$ of the corresponding symmetry axis is specified. In this setup, we obtain Le Cam optimal tests of uniformity, that are based on the sample covariance matrix (unlike their non-axial analogs, that are based on the sample average). For the more important unspecified-$\boldsymbol{\theta}$ problem, some classical tests are available in the literature, but virtually nothing is known on their non-null behavior. We therefore study the non-null behavior of the celebrated Bingham test and of other tests that exploit the single-spiked nature of the considered alternatives. We perform Monte Carlo exercises to investigate the finite-sample behavior of our tests and to show their agreement with our asymptotic results.

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Christine Cutting. Davy Paindaveine. Thomas Verdebout. "On the power of axial tests of uniformity on spheres." Electron. J. Statist. 14 (1) 2123 - 2154, 2020. https://doi.org/10.1214/20-EJS1716

Information

Received: 1 November 2019; Published: 2020
First available in Project Euclid: 13 May 2020

zbMATH: 07210997
MathSciNet: MR4097051
Digital Object Identifier: 10.1214/20-EJS1716

Subjects:
Primary: 62F05 , 62H11
Secondary: 62E20

Keywords: axial data , contiguity , directional statistics , local asymptotic normality , rotational symmetry , tests of uniformity

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Vol.14 • No. 1 • 2020
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