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

Abstract

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 a 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 the parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of Möbius transformation. Properties of a three-parameter symmetric submodel are investigated as well; these include maximum likelihood estimation, its asymptotics, and a reparameterization that proves useful quite generally. A three-parameter asymmetric subfamily, which often proves to be an adequate model, is also discussed, with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed data set and are then adopted as the angular error distribution of a circular–circular regression model. Two applications of the latter are given. It is in this regression context that the Möbius transformation especially comes into its own. Comparisons with other families of circular distributions are made. Supplemental materials for this article are available online.

Keywords:  :

• Asymmetric distribution
• Circular skewness
• Circular–circular regression
• Unimodality
• Von Mises distribution
• Wrapped Cauchy distribution