A Family of Symmetric Distributions on the Circle

Abstract

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.

Symmetric Unimodal Models for Directional Data Motivated by Inverse Stereographic Projection

“…Its density results on combining the defining structure of the pdf in (1.8), derived by inverse stereographic projection, with the transformation technique based on the hyperbolic tangent function employed in the construction of Jones and Pewsey (2005). The resulting density is given by…”

“…As the papers of Shimizu and Iida (2002), Minh and Farnum (2003) and Jones and Pewsey (2005) attest, recent years have seen renewed interest in the development of flexible models for directional data distributed on the unit circle or sphere. Historically, four general approaches have been used to generate circular distributions.…”

“…The three-parameter family of symmetric circular distributions proposed by Jones and Pewsey (2005) has density f (θ) = (cosh(κψ)) 1/ψ (1 + tanh(κψ) cos(θ − µ)) 1/ψ 2πP 1/ψ (cosh(κψ)) , (1.4) where −π ≤ µ < π is a location parameter (equal, in general, to the mean/modal/median direction), κ ≥ 0 is a concentration parameter, ψ ∈ R is a shape index, and P 1/ψ is the associated Legendre function of the first kind of degree 1/ψ and order 0. The family with density (1.4) is an important one because it includes most of the standard symmetric circular distributions.…”

Kernel density estimation on the torus

“…Unimodal symmetric distributions in the family of Kato and Jones (2009). 4. The family of unimodal symmetric distributions of Jones and Pewsey (2005).…”

A Projected Exponential Family for Modeling Semicircular Data

“…Even recent models appearing in Jones and Pewsey (2005), Pewsey et al (2007) are symmetric. Pewsey (2002Pewsey ( , 2004 considers the testing of problems where the underlying distribution is reflectively symmetric about an unknown central direction and about a median axis, respectively.…”