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

Understand

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.