The Wrapped Cauchy Distribution on the Circle and Related Statistical Models

Abstract

The wrapped Cauchy distribution or the circular Cauchy distribution is a probability distribution defined on the circle. In this paper we discuss two topics related to the wrapped Cauchy distribution. First some known results about this distribution are introduced, including basic properties, derivations, parameter estimation, association with the Möbius transformation, and comparison with the von Mises distribution. Second we provide a review of statistical models related to the wrapped Cauchy. In particular, the most of the second topic is devoted to introduce a bivariate extension of the wrapped Cauchy distribution proposed by Kato and Pewsey (2015). It is seen that this distribution has numerous tractable properties in terms of probability density function, interpretation of the parameters, marginals and conditionals, correlation coefficients, and parameter estimation.