The Open UniversitySkip to content

A family of distributions on the circle with links to, and applications arising From, Möbius Transformation

Kato, Shogo and Jones, M. C. (2010). A family of distributions on the circle with links to, and applications arising From, Möbius Transformation. Journal of the American Statistical Association, 105(489) pp. 249–262.

DOI (Digital Object Identifier) Link:
Google Scholar: Look up in Google Scholar


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

Item Type: Journal Item
Copyright Holders: 2010 American Statistical Association
ISSN: 1537-274X
Keywords: asymmetric distribution; circular skewness; circular–circular regression; unimodality; Von Mises distribution; wrapped Cauchy distribution.
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Related URLs:
Item ID: 25815
Depositing User: Sarah Frain
Date Deposited: 28 Dec 2010 22:44
Last Modified: 07 Dec 2018 09:46
Share this page:


Altmetrics from Altmetric

Citations from Dimensions

Actions (login may be required)

Policies | Disclaimer

© The Open University   contact the OU