The Annals of Statistics

An Antipodally Symmetric Distribution on the Sphere

Christopher Bingham

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Abstract

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.

Article information

Source
Ann. Statist., Volume 2, Number 6 (1974), 1201-1225.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342874

Digital Object Identifier
doi:10.1214/aos/1176342874

Mathematical Reviews number (MathSciNet)
MR397988

Zentralblatt MATH identifier
0297.62010

JSTOR
links.jstor.org

Subjects
Primary: 62E99: None of the above, but in this section
Secondary: 33A30 62F05: Asymptotic properties of tests 62F10: Point estimation

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

Citation

Bingham, Christopher. An Antipodally Symmetric Distribution on the Sphere. Ann. Statist. 2 (1974), no. 6, 1201--1225. doi:10.1214/aos/1176342874. https://projecteuclid.org/euclid.aos/1176342874


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