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arXiv · 1510.07679

M\"obius transformation and a Cauchy family on the sphere

Shogo Kato·Peter McCullagh
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Abstract

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 M\"obius transformation on the sphere and there is a similar induced transformation on the parameter space. Stereographic projection transforms the the spherical Cauchy family into a multivariate $t$-family with a certain degree of freedom on Euclidean space. Many tractable properties of the spherical Cauchy are derived using the M\"obius transformation and stereographic projection. A method of moments estimator and an asymptotically efficient estimator are expressed in closed form. The maximum likelihood estimation is also straightforward.

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Shogo Kato, Peter McCullagh. 2015-10-26. M\"obius transformation and a Cauchy family on the sphere. https://doi.org/10.3150/20-bej1222

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