An extended family of circular distributions related to wrapped Cauchy distributions via Brownian motion

TL;DR

This paper introduces a four-parameter family of circular distributions extending the wrapped Cauchy, derived via Brownian motion, with closed-form densities, tractable properties, and potential generalizations to spherical distributions.

Contribution

It presents a novel four-parameter family of circular distributions related to the wrapped Cauchy, derived through Brownian motion, with explicit formulas and broader applicability.

Findings

Closed-form density expressions for the new family

Trigonometric moments with simple forms

Potential for symmetric and asymmetric models

Abstract

We introduce a four-parameter extended family of distributions related to the wrapped Cauchy distribution on the circle. The proposed family can be derived by altering the settings of a problem in Brownian motion which generates the wrapped Cauchy. The densities of this family have a closed form and can be symmetric or asymmetric depending on the choice of the parameters. Trigonometric moments are available, and they are shown to have a simple form. Further tractable properties of the model are obtained, many by utilizing the trigonometric moments. Other topics related to the model, including alternative derivations and M\"{o}bius transformation, are considered. Discussion of the symmetric submodels is given. Finally, generalization to a family of distributions on the sphere is briefly made.