A family of symmetric distributions on the circle

We propose a new family of symmetric unimodal distributions on the circle that contains the uniform, von Mises, cardioid, and wrapped Cauchy distributions, among others, as special cases. The basic form of the densities of this family is very simple, although its normalization constant involves an associated Legendre function. The family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle. Trigonometric moments are available, and a measure of variation is discussed. Aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to an example set of data. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made.