A Family of Distributions on the Circle With Links to, and Applications Arising From, Möbius Transformation

We propose a family of four-parameter distributions on the circle that contains the von Mises and wrapped Cauchy distributions as special cases. The family is derived by transforming the von Mises distribution via Möbius transformation which maps the unit circle onto itself. The densities in the family have a symmetric or asymmetric, unimodal or bimodal shape, depending on the values of parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of the Möbius transformation. Properties of a three-parameter symmetric submodel are also investigated; these include maximum likelihood estimation, its asymptotics and a reparametrisation that proves useful quite generally. A three-parameter asymmetric subfamily, which proves adequate in each of the examples in the paper, is also discussed with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed dataset and are then adopted as the angular error distribution of a circular–circular regression model, and an application given thereof. It is in this regression context that the Möbius transformation particularly comes into its own. Comparisons with other families of circular distributions are made.