## The Annals of Statistics

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

### An Antipodally Symmetric Distribution on the Sphere

#### 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