Conformal mapping for multivariate Cauchy families

We discuss some statistical properties of the multivariate Cauchy families on the Euclidean space and on the sphere. It is seen that the two multivariate Cauchy families are closed under conformal mapping called the M\"obius transformation and that, for each Cauchy family, there is a similar induced transformation on the parameter space. Some properties of a marginal distribution of the spherical Cauchy such as certain moments and a closure property associated with the real M\"obius group are obtained. It is shown that the two multivariate Cauchy families are connected via stereographic projection. Maximum likelihood estimation for the two Cauchy families is considered; closed-form expressions for the maximum likelihood estimators are available when the sample size is not greater than three, and the unimodality holds for the maximized likelihood functions. A Kent-type extension of the spherical Cauchy arising from an extended M\"obius subgroup is briefly considered.