Möbius transformation and a Cauchy family on the sphere

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.