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November, 1974 An Antipodally Symmetric Distribution on the Sphere
Christopher Bingham
Ann. Statist. 2(6): 1201-1225 (November, 1974). DOI: 10.1214/aos/1176342874


The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and their asymptotic properties elucidated. Inference procedures, including tests for isotropy and circular symmetry, are proposed. The application of $\Psi$ is illustrated by a numerical example.


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Christopher Bingham. "An Antipodally Symmetric Distribution on the Sphere." Ann. Statist. 2 (6) 1201 - 1225, November, 1974.


Published: November, 1974
First available in Project Euclid: 12 April 2007

zbMATH: 0297.62010
MathSciNet: MR397988
Digital Object Identifier: 10.1214/aos/1176342874

Primary: 62E99
Secondary: 33A30 , 62F05 , 62F10

Keywords: distribution of axes , distribution of directions , Distribution on sphere , hypergeometric functions , test of circularity , test of isotropy

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 6 • November, 1974
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