an:07256174
Zbl 1455.60032
Kato, Shogo; Mccullagh, Peter
Some properties of a Cauchy family on the sphere derived from the M??bius transformations
EN
Bernoulli 26, No. 4, 3224-3248 (2020).
00453701
2020
j
60E05 60D05 60E10 62E15
directional statistics; high dimensional data; stereographic projection; von Mises-Fisher distribution; wrapped Cauchy distribution
Summary: 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??bius transformations on the sphere and the parameter of the transformed family is expressed using extended M??bius 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 M??bius 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.