A Family of Symmetric Distributions on the Circle

We propose a new family of symmetric unimodal distributions on the circle which contains the uniform, von Mises, cardioid and wrapped Cauchy distributions, amongst others, as special cases. The basic form of the densities of this family is very simple, although its normalisation 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 circular variance and kurtosis are investigated in detail. The new family is compared with the wrapped normal and wrapped symmetric stable distributions. Theoretical aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to two example sets of data. The problem of the sample sizes necessary in practice to reliably discriminate between the von Mises, cardioid and wrapped Cauchy distributions is investigated. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made.