Directional data analysis using the spherical Cauchy and the Poisson kernel-based distribution

Abstract: In 2020, two novel distributions for the analysis of directional data were introduced: the spherical Cauchy distribution and the Poisson kernel-based distribution. This paper provides a detailed exploration of both distributions within various analytical frameworks. To enhance the practical utility of these distributions, alternative parametrizations that offer advantages in numerical stability and parameter estimation are presented, such as implementation of the Newton-Raphson algorithm for parameter estimation, while facilitating a more efficient and simplified approach in the regression framework. Additionally, a two-sample location test based on the log-likelihood ratio test is introduced. This test is designed to assess whether the location parameters of two populations can be assumed equal. The maximum likelihood discriminant analysis framework is developed for classification purposes, and finally, the problem of clustering directional data is addressed, by fitting finite mixtures of Spherical Cauchy or Poisson kernel-based distributions. Empirical validation is conducted through comprehensive simulation studies and real data applications, wherein the performance of the spherical Cauchy and Poisson kernel-based distributions is systematically compared.