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November 2020 Some properties of a Cauchy family on the sphere derived from the Möbius transformations
Shogo Kato, Peter McCullagh
Bernoulli 26(4): 3224-3248 (November 2020). DOI: 10.3150/20-BEJ1222


We present some properties of a Cauchy family of distributions on the sphere, which is a spherical extension of the wrapped Cauchy family on the circle. The spherical Cauchy family is closed under the Möbius transformations on the sphere and the parameter of the transformed family is expressed using extended Möbius transformations on the compactified Euclidean space. Stereographic projection transforms the spherical Cauchy family into a multivariate $t$-family with a certain degree of freedom on Euclidean space. The Möbius transformations and stereographic projection enable us to obtain some results related to the spherical Cauchy family such as an efficient algorithm for random variate generation, a simple form of pivotal statistic and straightforward calculation of probabilities of a region. A method of moments estimator and an asymptotically efficient estimator are expressed in closed form. Maximum likelihood estimation is also straightforward.


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Shogo Kato. Peter McCullagh. "Some properties of a Cauchy family on the sphere derived from the Möbius transformations." Bernoulli 26 (4) 3224 - 3248, November 2020.


Received: 1 February 2019; Revised: 1 March 2020; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256174
MathSciNet: MR4140543
Digital Object Identifier: 10.3150/20-BEJ1222

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability


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Vol.26 • No. 4 • November 2020
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