Circular and spherical projected Cauchy distributions

We propose a new family of projected distributions on the circle and the sphere, the circular and the spherical projected Cauchy distributions. We show that the wrapped Cauchy distribution is a special of the circular projected Cauchy distribution. Further, a generalization of the wrapped Cauchy distribution is proposed, which includes an extra parameter that improves the fit of the distribution. For the spherical case, the imposition of two conditions on the scatter matrix makes the distribution elliptically symmetric, which simplifies its analysis. The projected distributions have nice features, such as closed-form normalizing constant and straightforward random value generation. The parameters of the distributions can be estimated via maximum likelihood, and their bias will be assessed through numerical studies. The proposed distributions have been compared to existing models using real data sets, and are shown to provide a better fit. Therefore, the circular projected and spherical projected Cauchy distributions are promising alternatives for modeling circular and directional data.